08. Newton Methods for Nonlinear Problems Affine Invariance and Adaptive Algorithms [Deuflhard 2011-09-15]

Springer Series in
Computational
Mathematics
Editorial Board
R. Bank
R.L. Graham
J. Stoer
R. Varga
H. Yserentant
35
Peter Deuflhard
Newton Methods
for Nonlinear Problems
Affine Invariance and Adaptive Algorithms
With 49 Figures
123
Peter Deuflhard
Zuse Institute Berlin (ZIB)
Takustr. 7
14195 Berlin, Germany
and
Freie Universität Berlin
Dept. of Mathematics and Computer Science
deufl[email protected]
Mathematics Subject Classification (2000): 65-01, 65-02, 65F10, 65F20,
65H10, 65H20, 65J15, 65L10, 65L60, 65N30, 65N55, 65P30
ISSN 0179-3632
ISBN 978-3-540-21099-7 (hardcover)
e-ISBN 978-3-642-23899-4
ISBN 978-3-642-23898-7 (softcover)
DOI 10.1007/978-3-642-23899-4
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2011937965
© Springer-Verlag Berlin Heidelberg 2004, Corrected printing 2006, First softcover
printing 201 1.
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Preface
In 1970, my former academic teacher Roland Bulirsch gave an exercise to
his students, which indicated the fascinating invariance of the ordinary Newton method under general affine transformation. To my surprise, however,
nearly all global Newton algorithms used damping or continuation strategies based on residual norms, which evidently lacked affine invariance. Even
worse, nearly all convergence theorems appeared to be phrased in not affine
invariant terms, among them the classical Newton-Kantorovich and NewtonMysovskikh theorem. In fact, in those days it was common understanding
among numerical analysts that convergence theorems were only expected to
give qualitative insight, but not too much of quantitative advice for application, apart from toy problems.
This situation left me deeply unsatisfied, from the point of view of both mathematical aesthetics and algorithm design. Indeed, since my first academic
steps, my scientific guideline has been and still is that ‘good’ mathematical
theory should have a palpable influence on the construction of algorithms,
while ‘good’ algorithms should be as firmly as possible backed by a transparently underlying mathematical theory. Only on such a basis, algorithms will
be efficient enough to cope with the enormous difficulties of real life problems.
In 1972, I started to work along this line by constructing global Newton algorithms with affine invariant damping strategies [59]. Early companions on this
road were Hans-Georg Bock, Gerhard Heindl, and Tetsuro Yamamoto. Since
then, the tree of affine invariance has grown lustily, spreading out in many
branches of Newton-type methods. So the plan of a comprehensive treatise
on the subject arose naturally. Florian Potra, Ekkehard Sachs, and Andreas
Griewank gave highly valuable detailed advice. Around 1992, a manuscript
on the subject with a comparable working title had already swollen to 300
pages and been distributed among quite a number of colleagues who used it
in their lectures or as a basis for their research. Clearly, these colleagues put
screws on me to ‘finish’ that manuscript.
However, shortly after, new relevant aspects came up. In 1993, my former
coworker Andreas Hohmann introduced affine contravariance in his PhD
thesis [120] as a further coherent concept, especially useful in the context
of inexact Newton methods with GMRES as inner iterative solver. From then
vi
Preface
on, the former ‘affine invariance’ had to be renamed, more precisely, as affine
covariance. Once the door had been opened, two more concepts arose: in
1996, myself and Martin Weiser formulated affine conjugacy for convex optimization [84]; a few years later, I found affine similarity to be important for
steady state problems in dynamical systems. As a consequence, I decided to
rewrite the whole manuscript from scratch, with these four affine invariance
concepts representing the columns of a structural matrix, whose rows are the
various Newton and Gauss-Newton methods. A presentation of details of the
contents is postponed to the next section.
This book has two faces: the first one is that of a textbook addressing itself
to graduate students of mathematics and computational sciences, the second
one is that of a research monograph addressing itself to numerical analysts
and computational scientists working on the subject.
As a textbook, selected chapters may be useful in classes on Numerical Analysis, Nonlinear Optimization, Numerical ODEs, or Numerical PDEs. The
presentation is striving for structural simplicity, but not at the expense of
precision. It contains a lot of theorems and proofs, from affine invariant versions of the classical Newton-Kantorovich and Newton-Mysovskikh theorem
(with proofs simpler than the traditional ones) up to new convergence theorems that are the basis for advanced algorithms in large scale scientific computing. I confess that I did not work out all details of all proofs, if they were
folklore or if their structure appeared repeatedly. More elaboration on this
aspect would have unduly blown up the volume without adding enough value
for the construction of algorithms. However, I definitely made sure that each
section is self-contained to a reasonable extent. At the end of each chapter,
exercises are included. Web addresses for related software are given.
As a research monograph, the presentation (a) quite often goes into the depth
covering a large amount of otherwise unpublished material, (b) is open in
many directions of possible future research, some of which are explicitly indicated in the text. Even though the experienced reader will have no difficulties
in identifying further open topics, let me mention a few of them: There is no
complete coverage of all possible combinations of local and global, exact and
inexact Newton or Gauss-Newton methods in connection with continuation
methods—let alone of all their affine invariant realizations; in other words,
the above structural matrix is far from being full. Moreover, apart from convex optimization and constrained nonlinear least squares problems, general
optimization and optimal control is left out. Also not included are recent results on interior point methods as well as inverse problems in L2 , even though
affine invariance has just started to play a role in these fields.
Preface
vii
Generally speaking, finite dimensional problems and techniques dominate the
material presented here—however, with the declared intent that the finite
dimensional presentation should filter out promising paths into the infinite
dimensional part of the mathematical world. This intent is exemplified in
several sections, such as
• Section 6.2 on ODE initial value problems, where stiff problems are analyzed via a simplified Newton iteration in function space—replacing the
Picard iteration, which appears to be suitable only for nonstiff problems,
• Section 7.4.2 on ODE boundary value problems, where an adaptive multilevel collocation method is worked out on the basis of an inexact Newton
method in function space,
• Section 8.1 on asymptotic mesh independence, where finite and infinite
dimensional Newton sequences are synoptically compared, and
• Section 8.3 on elliptic PDE boundary value problems, where inexact Newton multilevel finite element methods are presented in detail.
The algorithmic paradigm, given in Section 1.2.3 and used all over the whole
book, will certainly be useful in a much wider context, far beyond Newton
methods.
Unfortunately, after having finished this book, I will probably lose all my
scientific friends, since I missed to quote exactly that part of their work that
should have been quoted by all means. I cannot but apologize in advance,
hoping that some of them will maintain their friendship nevertheless. In fact,
as the literature on Newton methods is virtually unlimited, I decided to not
even attempt to screen or pretend to have screened all the relevant literature,
but to restrict the references essentially to those books and papers that are
either intimately tied to affine invariance or have otherwise been taken as
direct input for the presentation herein. Even with this restriction the list is
still quite long.
At this point it is my pleasure to thank all those coworkers at ZIB, who have
particularly helped me with the preparation of this book. My first thanks
go to Rainer Roitzsch, without whose high motivation and deep TEX knowledge this book could never have appeared. My immediate next thanks go
to Erlinda Körnig and Sigrid Wacker for their always friendly cooperation
over the long time that the manuscript has grown. Moreover, I am grateful
to Ulrich Nowak, Andreas Hohmann, Martin Weiser, and Anton Schiela for
their intensive computational assistance and invaluable help in improving the
quality of the manuscript.
viii
Preface
Nearly last, but certainly not least, I wish to thank Harry Yserentant, Christian Lubich, Matthias Heinkenschloss, and a number of anonymous reviewers
for valuable comments on a former draft. My final thanks go to Martin Peters
from Springer for his enduring support.
Berlin, February 2004
Peter Deuflhard
Preface to Second Printing
The enjoyably fast acceptance of this monograph has made a second printing
necessary. Compared to the first one, only minor corrections and citation
updates have been made.
Berlin, November 2005
Peter Deuflhard
Table of Contents
1
Outline of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Newton-Raphson Method for Scalar Equations . . . . . . . . . . . . .
1.2 Newton’s Method for General Nonlinear Problems . . . . . . . . . .
1.2.1 Classical convergence theorems revisited . . . . . . . . . . . . .
1.2.2 Affine invariance and Lipschitz conditions . . . . . . . . . . .
1.2.3 The algorithmic paradigm . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 A Roadmap of Newton-type Methods . . . . . . . . . . . . . . . . . . . . .
1.4 Adaptive Inner Solvers for Inexact Newton Methods . . . . . . . .
1.4.1 Residual norm minimization: GMRES . . . . . . . . . . . . . . . . .
1.4.2 Energy norm minimization: PCG . . . . . . . . . . . . . . . . . . . .
1.4.3 Error norm minimization: CGNE . . . . . . . . . . . . . . . . . . . .
1.4.4 Error norm reduction: GBIT . . . . . . . . . . . . . . . . . . . . . . . .
1.4.5 Linear multigrid methods . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
11
11
13
20
21
26
28
30
32
35
38
40
Part I ALGEBRAIC EQUATIONS
2
Systems of Equations: Local Newton Methods . . . . . . . . . . . .
2.1 Error Oriented Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Ordinary Newton method . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Simplified Newton method . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Newton-like methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Broyden’s ‘good’ rank-1 updates . . . . . . . . . . . . . . . . . . .
2.1.5 Inexact Newton-ERR methods . . . . . . . . . . . . . . . . . . . . .
2.2 Residual Based Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Ordinary Newton method . . . . . . . . . . . . . . . . . . . . . . . . .
45
45
45
52
56
58
67
76
76
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2.2.2 Simplified Newton method . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Broyden’s ‘bad’ rank-1 updates . . . . . . . . . . . . . . . . . . . .
2.2.4 Inexact Newton-RES method . . . . . . . . . . . . . . . . . . . . . .
2.3 Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Ordinary Newton method . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Simplified Newton method . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Inexact Newton-PCG method . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
81
90
94
94
97
98
104
3
Systems of Equations: Global Newton Methods . . . . . . . . . . .
3.1 Globalization Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Componentwise convex mappings . . . . . . . . . . . . . . . . . . .
3.1.2 Steepest descent methods . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Trust region concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Newton path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Residual Based Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Affine contravariant convergence analysis . . . . . . . . . . . .
3.2.2 Adaptive trust region strategies . . . . . . . . . . . . . . . . . . . .
3.2.3 Inexact Newton-RES method . . . . . . . . . . . . . . . . . . . . . .
3.3 Error Oriented Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 General level functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Natural level function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Adaptive trust region strategies . . . . . . . . . . . . . . . . . . . .
3.3.4 Inexact Newton-ERR methods . . . . . . . . . . . . . . . . . . . . .
3.4 Convex Functional Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Affine conjugate convergence analysis . . . . . . . . . . . . . . .
3.4.2 Adaptive trust region strategies . . . . . . . . . . . . . . . . . . . .
3.4.3 Inexact Newton-PCG method . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
110
111
114
117
121
125
126
128
131
134
135
138
145
152
161
162
165
168
170
4
Least Squares Problems: Gauss-Newton Methods . . . . . . . . .
4.1 Linear Least Squares Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Unconstrained problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Equality constrained problems . . . . . . . . . . . . . . . . . . . . .
4.2 Residual Based Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Local Gauss-Newton methods . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Global Gauss-Newton methods . . . . . . . . . . . . . . . . . . . . .
173
175
175
178
182
183
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5
xi
4.2.3 Adaptive trust region strategy . . . . . . . . . . . . . . . . . . . . .
4.3 Error Oriented Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Local convergence results . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Local Gauss-Newton algorithms . . . . . . . . . . . . . . . . . . . .
4.3.3 Global convergence results . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Adaptive trust region strategies . . . . . . . . . . . . . . . . . . . .
4.3.5 Adaptive rank strategies . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Underdetermined Systems of Equations . . . . . . . . . . . . . . . . . . .
4.4.1 Local quasi-Gauss-Newton method . . . . . . . . . . . . . . . . .
4.4.2 Global Gauss-Newton method . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190
193
194
197
205
212
215
221
221
225
228
Parameter Dependent Systems: Continuation Methods . . .
5.1 Newton Continuation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Classification of continuation methods . . . . . . . . . . . . . .
5.1.2 Affine covariant feasible stepsizes . . . . . . . . . . . . . . . . . . .
5.1.3 Adaptive pathfollowing algorithms . . . . . . . . . . . . . . . . . .
5.2 Gauss-Newton Continuation Method . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Discrete tangent continuation beyond turning points . .
5.2.2 Affine covariant feasible stepsizes . . . . . . . . . . . . . . . . . . .
5.2.3 Adaptive stepsize control . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Computation of Simple Bifurcations . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Augmented systems for critical points . . . . . . . . . . . . . .
5.3.2 Newton-like algorithm for simple bifurcations . . . . . . .
5.3.3 Branching-off algorithm . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
233
237
237
242
245
250
250
253
257
263
263
271
277
280
Part II DIFFERENTIAL EQUATIONS
6
Stiff ODE Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Affine Similar Linear Contractivity . . . . . . . . . . . . . . . . . . . . . . .
6.2 Nonstiff versus Stiff Initial Value Problems . . . . . . . . . . . . . . . .
6.2.1 Picard iteration versus Newton iteration . . . . . . . . . . . . .
6.2.2 Newton-type uniqueness theorems . . . . . . . . . . . . . . . . . .
6.3 Uniqueness Theorems for Implicit One-step Methods . . . . . . . .
6.4 Pseudo-transient Continuation for Steady State Problems . . . .
285
285
288
288
291
295
299
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6.4.1 Exact pseudo-transient continuation . . . . . . . . . . . . . . . . 300
6.4.2 Inexact pseudo-transient continuation . . . . . . . . . . . . . . . 307
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
7
ODE Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Multiple Shooting for Timelike BVPs . . . . . . . . . . . . . . . . . . . . .
7.1.1 Cyclic linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Realization of Newton methods . . . . . . . . . . . . . . . . . . . .
7.1.3 Realization of continuation methods . . . . . . . . . . . . . . . .
7.2 Parameter Identification in ODEs . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Periodic Orbit Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Single orbit computation . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Orbit continuation methods . . . . . . . . . . . . . . . . . . . . . . .
7.3.3 Fourier collocation method . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Polynomial Collocation for Spacelike BVPs . . . . . . . . . . . . . . . .
7.4.1 Discrete versus continuous solutions . . . . . . . . . . . . . . . .
7.4.2 Quasilinearization as inexact Newton method . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
315
316
318
324
327
333
337
337
340
344
349
350
356
366
8
PDE Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Asymptotic Mesh Independence . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Global Discrete Newton Methods . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 General PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2 Elliptic PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs . . . . . . . . . .
8.3.1 Local Newton-Galerkin methods . . . . . . . . . . . . . . . . . . . .
8.3.2 Global Newton-Galerkin methods . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
369
369
378
378
385
389
391
397
403
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
Outline of Contents
This book is divided into eight chapters, a reference list, a software list, and
an index. After an elementary introduction in Chapter 1, it splits into two
parts: Part I, Chapter 2 to Chapter 5, on finite dimensional Newton methods
for algebraic equations, and Part II, Chapter 6 to Chapter 8, on extensions
to ordinary and partial differential equations. Exercises are added at the end
of each chapter.
Chapter 1. This introductory chapter starts from the historical root, Newton’s method for scalar equations (Section 1.1). The method can be derived
either algebraically, which leads to local Newton methods only (presented in
Chapter 2), or geometrically, which leads to global Newton methods via the
concept of the Newton path (see Chapter 3).
The next Section 1.2 contains the key to the basic understanding of this monograph. First, four affine invariance classes are worked out, which represent the
four basic strands of this treatise:
•
•
•
•
affine
affine
affine
affine
covariance, which leads to error norm controlled algorithms,
contravariance, which leads to residual norm controlled algorithms,
conjugacy, which leads to energy norm controlled algorithms, and
similarity, which may lead to time step controlled algorithms.
Second, the affine invariant local estimation of affine invariant Lipschitz constants is set as the central paradigm for the construction of adaptive Newton
algorithms.
In Section 1.3, we give a roadmap of the large variety of Newton-type
methods—essentially fixing terms to be used throughout the book such as ordinary and simplified Newton method, Newton-like methods, inexact Newton
methods, quasi-Newton methods, Gauss-Newton methods, quasilinearization,
or inexact Newton multilevel methods. In Section 1.4, we briefly collect details about iterative linear solvers to be used as inner iterations within finite
dimensional inexact Newton algorithms; each affine invariance class is linked
with a special class of inner iterations. In view of function space oriented inexact Newton algorithms, we also revisit linear multigrid methods. Throughout
this section, we emphasize the role of adaptive error control.
1
2
Outline
PART I. The following Chapters 2 to 5 deal with finite dimensional Newton
methods for algebraic equations.
Chapter 2. This chapter deals with local Newton methods for the numerical
solution of systems of nonlinear equations with finite, possibly large dimension. The term ‘local’ refers to the situation that ‘sufficiently good’ initial
guesses of the solution are assumed to be at hand. Special attention is paid
to the issue of how to recognize, whether a given initial guess x0 is ‘sufficiently good’. Different affine invariant formulations give different answers
to this question, in theoretical terms as well as by virtue of the algorithmic
paradigm of Section 1.2.3. Problems of this structure are called ‘mildly nonlinear’; their computational complexity can be bounded a-priori in units of
the computational complexity of the corresponding linearized system.
As it turns out, different affine invariant Lipschitz conditions, which have
been introduced in Section 1.2.2, lead to different characterizations of local
convergence domains in terms of error oriented norms, residual norms, or
energy norms, which, in turn, give rise to corresponding variants of Newton
algorithms. We give three different, strictly affine invariant convergence analyses for the cases of affine covariant (error oriented) Newton methods (Section 2.1), affine contravariant (residual based) Newton methods (Section 2.2),
and affine conjugate Newton methods for convex optimization (Section 2.3).
Details are worked out for ordinary Newton algorithms, simplified Newton algorithms, and inexact Newton algorithms—synoptically for each of the three
affine invariance classes. Moreover, affine covariance is naturally associated
with Broyden’s ‘good’ quasi-Newton method, whereas affine contravariance
corresponds to Broyden’s ‘bad’ quasi-Newton method.
Affine invariant globalization, which means global extension of the convergence domains of local Newton methods in the affine invariant frame, is possible along several lines:
• global Newton methods with damping strategy—see Chapter 3,
• parameter continuation methods—see Chapter 5,
• pseudo-transient continuation methods—see Section 6.4.
Chapter 3. This chapter deals with global Newton methods for systems of
nonlinear equations with finite, possibly large dimension. The term ‘global’
refers to the situation that here, in contrast to the preceding chapter, ‘sufficiently good’ initial guesses of the solution are no longer assumed. Problems
of this structure are called ‘highly nonlinear’; their computational complexity
depends on topological details of Newton paths associated with the nonlinear
mapping and can typically not be bounded a-priori.
Outline
3
In Section 3.1 we survey globalization concepts such as
•
•
•
•
steepest descent methods,
trust region methods,
the Levenberg-Marquardt method, and
the Newton method with damping strategy.
In Section 3.1.4, a rather general geometric approach is taken: the idea is
to derive a globalization concept without a pre-occupation to any iterative
method, just starting from the requirement of affine covariance as a ‘first
principle’. Surprisingly, this general approach leads to a topological derivation
of Newton’s method with damping strategy via Newton paths.
In order to accept or reject a new iterate, monotonicity tests are applied.
We study different such tests, according to different affine invariance requirements:
• the most popular residual monotonicity test, which is related to affine contravariance (Section 3.2),
• the error oriented so-called natural monotonicity test, which is related to
affine covariance (Section 3.3), and
• the convex functional test as the natural requirement in convex optimization, which reflects affine conjugacy (Section 3.4).
For each of these three affine invariance classes, adaptive trust region strategies are designed in view of an efficient choice of damping factors in Newton’s
method. They are all based on the paradigm of Section 1.2.3. On a theoretical
basis, details of algorithmic realization in combination with either direct or
iterative linear solvers are worked out. As it turns out, an efficient determination of the steplength factor in global inexact Newton methods is intimately
linked with the accuracy matching for affine invariant combinations of inner
and outer iteration.
Chapter 4. This chapter deals with both local and global Gauss-Newton
methods for nonlinear least squares problems in finite dimension—a method,
which attacks the solution of the nonlinear least squares problem by solving a
sequence of linear least squares problems. Affine invariance of both theory and
algorithms will once again play a role, here restricted to affine contravariance
and affine covariance. The theoretical treatment requires considerably more
sophistication than in the simpler case of Newton methods for nonlinear
equations.
In order to lay some basis, unconstrained and equality constrained linear
least squares problems are first discussed in Section 4.1, introducing the useful calculus of generalized inverses. In Section 4.2, an affine contravariant
convergence analysis of Gauss-Newton methods is given and worked out in
the direction of residual based algorithms. Local convergence turns out to
4
Outline
be only guaranteed for ‘small residual’ problems, which can be characterized
in theoretical and algorithmic terms. Local and global convergence analysis
as well as adaptive trust region strategies rely on some projected residual
monotonicity test. Both unconstrained and separable nonlinear least squares
problems are treated.
In the following Section 4.3, local convergence of error oriented Gauss-Newton
methods is studied in affine covariant terms; again, Gauss-Newton methods
are seen to exhibit guaranteed convergence only for a restricted problem class,
named ‘adequate’ nonlinear least squares problems, since they are seen to be
adequate in terms of the underlying statistical problem formulation. The
globalization of these methods is done via the construction of two topological
paths: the local and the global Gauss-Newton path. In the special case of
nonlinear equations, the two paths coincide to one path, the Newton path.
On this theoretical basis, adaptive trust region strategies (including rank
strategies) combined with a natural extension of the natural monotonicity
test are presented in detail for unconstrained, for separable, and—in contrast
to the residual based approach—also for nonlinearly constrained nonlinear
least squares problems. Finally, in Section 4.4, we study underdetermined
nonlinear systems. In this case, a geodetic Gauss-Newton path exists generically and can be exploited to construct a quasi-Gauss-Newton algorithm and
a corresponding adaptive trust region method.
Chapter 5. This chapter discusses the numerical solution of parameter dependent systems of nonlinear equations, which is the basis for parameter
studies in systems analysis and systems design as well as for the globalization of local Newton methods. The key concept behind the approach is the
(possible) existence of a homotopy path with respect to the selected parameter. In order to follow such a path, we here advocate discrete continuation
methods, which consist of two essential parts:
• a prediction method, which, from given points on the homotopy path, produces some ‘new’ point assumed to be ‘sufficiently close’ to the homotopy
path,
• an iterative correction method, which, from a given starting point close to,
but not on the homotopy path, supplies some point on the homotopy path.
For the prediction step, classical or tangent continuation are the canonical
choices. Needless to say that, for the iterative correction steps, we here concentrate on local Newton and (underdetermined) Gauss-Newton methods.
Since the homotopy path is a mathematical object in the domain space of
the nonlinear mapping, we only present the affine covariant approach.
In Section 5.1, we derive an adaptive Newton continuation algorithm with
the ordinary Newton method as correction; this algorithm terminates locally
in the presence of critical points including turning points. In order to follow
the path beyond turning points, a quasi-Gauss-Newton continuation algo-
Outline
5
rithm is worked out in Section 5.2, based on the preceding Section 4.4. This
algorithm still terminates in the neighborhood of any higher order critical
point. In order to overcome such points as well, we exemplify a scheme to
construct augmented systems, whose solutions are just selected critical points
of higher order—see Section 5.3. This scheme is an appropriate combination
of Lyapunov-Schmidt reduction and topological universal unfolding. Details
of numerical realization are only worked out for the computation of diagrams
including simple bifurcation points.
PART II. The following Chapters 6 to 8 deal predominantly with infinite
dimensional , i.e., function space oriented Newton methods. The selected topics are stiff initial value problems for ordinary differential equations (ODEs)
and boundary value problems for ordinary and partial differential equations
(PDEs).
Chapter 6. This chapter deals with stiff initial value problems for ODEs.
The discretization of such problems is known to involve the solution of nonlinear systems per each discretization step—in one way or the other.
In Section 6.1, the contractivity theory for linear ODEs is revisited in terms
of affine similarity. Based on an affine similar convergence theory for a simplified Newton method in function space, a nonlinear contractivity theory for
stiff ODE problems is derived in Section 6.2, which is quite different from
the theory given in usual textbooks on the topic. The key idea is to replace
the Picard iteration in function space, known as a tool to show uniqueness in
nonstiff initial value problems, by a simplified Newton iteration in function
space to characterize stiff initial value problems. From this point of view, linearly implicit one-step methods appear as direct realizations of the simplified
Newton iteration in function space. In Section 6.3, exactly the same theoretical characterization is shown to apply also to implicit one-step methods,
which require the solution of a nonlinear system by some finite dimensional
Newton-type method at each discretization step.
Finally, in a deliberately longer Section 6.4, we discuss pseudo-transient continuation algorithms, whereby steady state problems are solved via stiff integration. This type of algorithm is particularly useful, when the Jacobian
matrix is singular due to hidden dynamical invariants (such as mass conservation). The (nearly) affine similar theoretical characterization permits
the derivation of an adaptive (pseudo-)time step strategy and an accuracy
matching strategy for a residual based inexact variant of the algorithm.
Chapter 7. In this chapter, we consider nonlinear two-point boundary value
problems for ODEs. The presentation and notation is closely related to Chapter 8 in the textbook [71]. Algorithms for the solution of such problems can be
grouped into two approaches: initial value methods such as multiple shooting
and global discretization methods such as collocation. Historically, affine covariant Newton methods have first been applied to this problem class—with
significant success.
6
Outline
In Section 7.1, the realization of Newton and discrete continuation methods
within the standard multiple shooting approach is elaborated. Gauss-Newton
methods for parameter identification in ODEs are discussed in Section 7.2,
also based on multiple shooting. For periodic orbit computation, Section 7.3
presents Gauss-Newton methods, both in the shooting approach (Sections
7.3.1 and 7.3.2) and in a Fourier collocation approach, also called Urabe or
harmonic balance method (Section 7.3.3).
In Section 7.4 we concentrate on polynomial collocation methods, which have
reached a rather mature status including affine covariant Newton methods. In
Section 7.4.1, the possible discrepancy between discrete and continuous solutions is studied including the possible occurrence of so-called ‘ghost solutions’
in the nonlinear case. On this basis, the realization of quasilinearization is
seen to be preferable in combination with collocation. The following Section
7.4.2 is then devoted to the key issue that quasilinearization can be interpreted as an inexact Newton method in function space: the approximation
errors in the infinite dimensional setting just replace the inner iteration errors arising in the finite dimensional setting. With this insight, an adaptive
multilevel control of the collocation errors can be realized to yield an adaptive
inexact Newton method in function space—which is the bridge to adaptive
Newton multilevel methods for PDEs (compare Section 8.3).
Chapter 8. This chapter deals with Newton methods for boundary value
problems in nonlinear PDEs. There are two principal approaches: (a) finite
dimensional Newton methods applied to a given system of already discretized
PDEs, also called discrete Newton methods, and (b) function space oriented
Newton methods applied to the continuous PDEs, at best in the form of
inexact Newton multilevel methods.
Before we discuss the two principal approaches in detail, we present an affine
covariant analysis of asymptotic mesh independence that connects the finite
dimensional and the infinite dimensional Newton methods, see Section 8.1.
In Section 8.2, we assume the standard situation in industrial technology
software, where the grid generation module is strictly separated from the
solution module. Consequently, nonlinear PDEs arise there as discrete systems of nonlinear equations with fixed finite, but usually high dimension and
large sparse ill-conditioned Jacobian matrix. This is the domain of applicability of finite dimensional inexact Newton methods. More advanced, but often
less favored in the huge industrial software environments, are function space
oriented inexact Newton methods, which additionally include the adaptive
manipulation of discretization meshes within a multilevel or multigrid solution process. This situation is treated in Section 8.3 and compared there with
finite dimensional inexact Newton techniques.
1 Introduction
This chapter is an elementary introduction into the general theme of this
book. We start from the historical root, Newton’s method for scalar equations
(Section 1.1): the method can be derived either algebraically, which leads to
local Newton methods only (see Chapter 2), or geometrically, which leads to
global Newton methods via the topological Newton path (see Chapter 3).
Section 1.2 contains the key to the basic understanding of this monograph.
First, four affine invariance classes are worked out, which represent the four
basic strands of this treatise:
•
•
•
•
affine
affine
affine
affine
covariance, which leads to error norm controlled algorithms,
contravariance, which leads to residual norm controlled algorithms,
conjugacy, which leads to energy norm controlled algorithms, and
similarity, which may lead to time step controlled algorithms.
Second, the affine invariant local estimation of affine invariant Lipschitz constants is set as the central paradigm for the construction of adaptive Newton
algorithms.
In Section 1.3, we fix terms for various Newton-type methods to be named
throughout the book: ordinary and simplified Newton method, Newton-like
methods, inexact Newton methods, quasi-Newton methods, quasilinearization, and inexact Newton multilevel methods.
In Section 1.4, details are given for the iterative linear solvers GMRES, PCG,
CGNE, and GBIT to an extent necessary to match them with finite dimensional
inexact Newton algorithms. In view of function space oriented inexact Newton
algorithms, we also revisit multiplicative, additive, and cascadic multigrid
methods emphasizing the role of adaptive error control therein.
1.1 Newton-Raphson Method for Scalar Equations
Assume we have to solve the scalar equation
f (x) = 0
with an appropriate guess x of the unknown solution x∗ at hand.
0
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance
and Adaptive Algorithms, Springer Series in Computational Mathematics 35,
DOI 10.1007/978-3-642-23899-4_1, © Springer-Verlag Berlin Heidelberg 2011
7
8
1 Introduction
Algebraic approach. We use the perturbation
Δx = x∗ − x0
for Taylor’s expansion
0 = f (x0 + Δx) = f (x0 ) + f (x0 )Δx + O(|Δx|2 ).
Upon dropping terms of order higher than linear in the perturbation, we
arrive at the approximate equation
f (x0 )Δx ≈ −f (x0 ),
which, assuming f (x0 ) = 0, leads to the precise equation
x1 − x0 = Δx0 = −
f (x0 )
f (x0 )
for a first correction of the starting guess. From this, an iterative scheme is
constructed by repetition
xk+1 = Φ(xk ) = xk −
f (xk )
, k = 0, 1, . . . .
f (xk )
If we study the contraction mapping Φ in terms of a contraction factor Θ, we
arrive at
f (x)f (x)
Θ = max Φ (x) = max
x∈I
x∈I
(f (x))2
with I an appropriate interval containing x∗ . From this, we have at least
linear convergence
|xk+1 − x∗ | ≤ Θ|xk − x∗ |
in a neighborhood of x∗ , where Θ < 1. In passing we note that this contraction
factor Θ remains unchanged, if we rescale the equation according to
αf (βy) = 0 , αβ = 0 , x = βy .
An extension of this kind of observation to rather general nonlinear problems
will lead to fruitful theoretical and algorithmical consequences below. For
starting guesses x0 ‘sufficiently close’ to x∗ even quadratic convergence of the
iterates can be shown in the sense that
|xk+1 − x∗ | ≤ C|xk − x∗ |2 , k = 0, 1, 2 . . . .
The algebraic derivation in terms of the linear perturbation treatment carries
over to rather general nonlinear problems up to operator equations such as
boundary value problems for ordinary or partial differential equations.
1.1 Newton-Raphson Method for Scalar Equations
9
Geometric approach. Looking at the graph of f (x)—as depicted in Figure 1.1—any root can be interpreted as the intersection of this graph with the
real axis. Since this intersection cannot be constructed other than by tedious
sampling of f , the graph of f (x) is replaced by its tangent p(x) in x0 and the
first iterate x1 is defined as the intersection of the tangent with the real axis.
Upon repeating this geometric process, the close-by solution point x∗ can be
constructed up to any desired accuracy. By geometric insight, the iterative
process will converge globally for convex (or concave) f —which includes the
case of arbitrarily ‘bad’ initial guesses as well! At first glance, this geometric
derivation seems to be restricted to the scalar case, since the graph of f (x)
is a typically one-dimensional concept. A careful examination of the subject
in more than one dimension, however, naturally leads to a topological path
called Newton path—see Section 3.1.4 below.
f
0
f (x0 )
x0
x∗
x
Fig. 1.1. Geometric interpretation: Newton’s method for a scalar equation.
Historical Note. Strictly speaking, Newton’s method could as well be
named as Newton-Raphson-Simpson method—as elaborated in recent articles by N. Kollerstrom [134] or T.J. Ypma [203]. According to these careful
historical studies, the following facts seem to be agreed upon among the
experts:
• In the year 1600, Francois Vieta (1540–1603) had (first?) designed a perturbation technique for the solution of the scalar polynomial equations,
which supplied one decimal place of the unknown solution per step via the
explicit calculation of successive polynomials of the successive perturbations. It seems that this method had also been detected independently by
al-Kāshı̄ and simplified around 1647 by Oughtred.
• Isaac Newton (1643–1727) got to know Vieta’s method in 1664. Up to
1669 he had improved it by linearizing these successive polynomials. As an
example, he discussed the numerical solution of the cubic polynomial
f (x) := x3 − 2x − 5 = 0 .
10
1 Introduction
Newton first noted that the integer part of the root is 2 setting x0 = 2.
Next, by means of x = 2 + p, he obtained the polynomial equation
p3 + 6p2 + 10p − 1 = 0 .
Herein he neglected terms higher than first order and thus put p ≈ 0.1. He
inserted p = 0.1 + q and constructed the polynomial equation
q 3 + 6.3q 2 + 11.23q + 0.061 = 0 .
Again he neglected terms higher than linear and found q ≈ −0.0054. Continuation of the process one more step led him to r ≈ 0.00004853 and
therefore to the third iterate
x3 = x0 + p + q + r = 2.09455147 .
Note that the relations 10p − 1 = 0 and 11.23q + 0.061 = 0 given above
correspond precisely to
p = x1 − x0 = −f (x0 )/f (x0 )
and to
q = x2 − x1 = −f (x1 )/f (x1 ) .
As the example shows, he had also observed that by keeping all decimal places of the corrections, the number of accurate places would double
per each step—i.e., quadratic convergence. In 1687 (Philosophiae Naturalis
Principia Mathematica), the first nonpolynomial equation showed up: it is
the well-known equation from astronomy
x − e sin(x) = M
between the mean anomaly M and the eccentric anomaly x. Here Newton
used his already developed polynomial techniques via the series expansion
of sin and cos. However, no hint on the derivative concept is incorporated!
• In 1690, Joseph Raphson (1648–1715) managed to avoid the tedious computation of the successive polynomials, playing the computational scheme
back to the original polynomial; in this now fully iterative scheme, he
also kept all decimal places of the corrections. He had the feeling that
his method differed from Newton’s method at least by its derivation.
• In 1740, Thomas Simpson (1710–1761) actually introduced derivatives
(‘fluxiones’) in his book ‘Essays on Several Curious and Useful Subjects
in Speculative and Mix’d Mathematicks, Illustrated by a Variety of Examples’. He wrote down the true iteration for one (nonpolynomial) equation
and for a system of two equations in two unknowns thus making the correct
extension to systems for the first time. His notation is already quite close
to our present one (which seems to go back to J. Fourier).
1.2 Newton’s Method for General Nonlinear Problems
11
Throughout this book, we will use the name ‘Newton-Raphson method’ only
for scalar equations. For general equations we will use the name ‘Newton
method’—even though the name ‘Newton-Simpson method’ would be more
appropriate in view of the just described historical background.
1.2 Newton’s Method for General Nonlinear Problems
In contrast to the preceding section, we now approach the general case. Assume we have to solve a nonlinear operator equation
F (x) = 0 ,
wherein F : D ⊂ X → Y for Banach spaces X, Y endowed with norms
· X and · Y . Let F be at least once continuously differentiable. Suppose
we have a starting guess x0 of the unknown solutions x∗ at hand. Then
successive linearization leads to the general Newton method
F (xk )Δxk = −F (xk ) , xk+1 = xk + Δxk ,
k = 0, 1, . . . .
(1.1)
Obviously, this method attacks the solution of a nonlinear problem by solving
a sequence of linear problems of the same kind.
1.2.1 Classical convergence theorems revisited
A necessary assumption for the solvability of the above linear problems is
that the derivatives F (x) are invertible for all occurring arguments. For this
reason, standard convergence theorems typically require a-priori that the
inverse F (x)−1 exists and is bounded
F (x)−1 Y →X ≤ β < ∞ , x ∈ D ,
(1.2)
where · Y →X denotes an operator norm. From a computational point of
view, such a theoretical quantity β defined over the domain D seems to be
hard to get, apart from rather simple examples. Sampling of local estimates
like
F (x0 )−1 Y →X ≤ β0
(1.3)
seems to be preferable, but is still quite expensive. Moreover, a well-known
rule in Numerical Analysis states that the actual computation of inverses
should be avoided. Rather, such a condition should be monitored implicitly
in the course of solving linear systems with specific right hand sides.
In order to study the convergence properties of the above Newton iteration,
some second derivative information is needed, as already stated in the scalar
equation case (Section 1.1 above). The classical standard form to include this
information is via a Lipschitz condition of the type
12
1 Introduction
F (x) − F (x̄)X→Y ≤ γx − x̄X , x, x̄ ∈ D .
(1.4)
With this additional assumption, the operator perturbation lemma (sometimes also called Banach perturbation lemma) proves the existence of some
upper bound β such that
F (x)−1 Y →X ≤ β ≤
for
x − x0 X <
β0
1 − β0 γx − x0 X
1
, x ∈ D.
β0 γ
The proof is left as Exercise 1.1. Classical convergence theorems for Newton’s
method use certain combinations of these assumptions.
Newton-Kantorovich theorem. This first classical convergence theorem
for Newton’s method in abstract spaces (see [127, 163]) requires assumptions
(1.3) and (1.4) to show existence and uniqueness of a solution x∗ as well as
quadratic convergence of the Newton iterates within a neighborhood characterized by a so-called Kantorovich quantity
h0 := Δx0 X β0 γ <
1
2
and a corresponding convergence ball around x0 with radius ρ0 ∼ 1/β0 γ.
This theorem is also the standard tool to prove the classical implicit function
theorem—compare Exercise 1.2.
Newton-Mysovskikh theorem. This second classical convergence theorem (see [155, 163]) requires assumptions (1.2) and (1.4) to show uniqueness
(not existence!) and quadratic convergence within a neighborhood characterized by the slightly different quantity
h0 := Δx0 X βγ < 2
and a corresponding convergence ball around x0 with radius ρ ∼ 1/βγ.
Both theorems seem to require the actual computation of the Lipschitz constant γ. However, such a quantity is certainly hard if not hopeless to compute
in realistic nonlinear problems. Moreover, even computational local estimates
of β and γ are typically far off any use in practical applications. That is why,
for quite a time, people believed that convergence results are of theoretical
interest only, but not of any value for the actual implementation of Newton
algorithms. An illustrating simple example is given as Exercise 2.3.
This undesirable gap between convergence analysis and algorithm construction has been the motivation for the present book. As will become apparent,
the key to closing this gap is supplied by affine invariance in both convergence
theory and algorithmic realization.
1.2 Newton’s Method for General Nonlinear Problems
13
1.2.2 Affine invariance and Lipschitz conditions
In order to make the essential point clear enough, it is sufficient to regard
simply systems of nonlinear equations, which means that X = Y = Rn for
fixed dimension n > 1 and the same norm in X and Y . Recall Newton’s
method in the form
F (xk )Δxk = −F (xk ), xk+1 = xk + Δxk
k = 0, 1, . . . .
Scaling. In sufficiently complex problems, scaling or re-gauging of variables
(say, from km to miles) needs to be carefully considered. Formally speaking,
with preselected nonsingular diagonal scaling matrices DL , DR for left and
right scaling, we may write
−1
(DL F (xk )DR )(DR
Δxk ) = −DL F (xk )
for the scaled linear system. Despite its formal equivalence with (1.1), all
standard norms used in Newton algorithms must now be replaced by scaled
norms such that (dropping the iteration index k)
−1
Δx , F , F + F (x)Δx −→ DR
Δx , DL F , DL (F + F (x)Δx) .
With the change of norms comes a change of the criteria for the acceptance or
rejection of new iterates. The effect of scaling on the iterative performance of
Newton-type methods is a sheet lightning of the more general effects caused
by affine invariance, which are the topic of this book.
Affine transformation. Let A, B ∈ Rn×n be arbitrary nonsingular matrices and study the affine transformations of the nonlinear system as
G(y) = AF (By) = 0 , x = By .
Then Newton’s method applied to G(y) reads
G (y k )Δy k = −G(y k ), y k+1 = y k + Δy k
With the relation
k = 0, 1, . . . .
G (y k ) = AF (xk )B
and starting guess y 0 = B −1 x0 we immediately obtain
xk = By k , k = 0, 1, . . . .
Obviously, the iterates are invariant under transformation of the image space
(by A)—an invariance property described by affine covariance. Moreover,
they are transformed just as the whole original space (by B)—a property
denoted by affine contravariance.
It is only natural to require that the above affine invariance properties are
inherited by any theoretical characterization. As it turns out, the inheritance
of the full invariance property is impossible. That is why we restrict our study
to four special invariance classes.
14
1 Introduction
Affine covariance. In this setting, we keep the domain space of F fixed
(B = I) and look at the whole class of problems
G(x) = AF (x) = 0
that is generated by the class GL(n) of nonsingular matrices A. The Newton
iterates are the same all over the whole class of nonlinear problems. For this
reason, an affine covariant theory about their convergence must be possible.
Upon revisiting the above theoretical assumptions (1.2), (1.3), and (1.4) we
now obtain
G (x)−1 ≤ β(A) , G (x0 )−1 ≤ β0 (A) , G (x) − G (x̄) ≤ γ(A)x − x̄ .
Application of the classical convergence theorems then yields convergence
balls with radius, say
ρ(A) ∼ 1/β(A)γ(A) .
Compared with β(I), γ(I) we obtain (assuming best possible theoretical
bounds)
β(A) ≤ β(I)A−1 , γ(A) ≤ γ(I)A
and therefore
β(A)γ(A) ≤ β(I)γ(I) cond(A) .
(1.5)
For n > 1 we have cond(A) ≥ 1, even unbounded for A ∈ GL(n). Obviously,
by a mean choice of A we can make the classical convergence balls shrink to
nearly zero!
Fortunately, careful examination of the proof of the Newton-Kantorovich theorem shows that assumptions (1.3) and (1.4) can be telescoped to the requirement
F (x0 )−1 F (x) − F (x̄) ≤ ω0 x − x̄ , x, x̄, x0 ∈ D .
(1.6)
The thus defined Lipschitz constant ω0 is affine covariant, since
G (x0 )−1 G (x) − G (x̄)
−1 AF (x0 )
A F (x) − F (x̄)
= F (x0 )−1 F (x) − F (x̄)
=
so that both sides of (1.6) are independent of A. This definition of ω0 (assumed best possible) still has the disadvantage of containing an operator
norm on the left side—which, however, is unavoidable, because the operator
perturbation lemma is required in the proof. Examination of the NewtonMysovskikh theorem shows that assumptions (1.2) and (1.4) can also be
telescoped to an affine covariant Lipschitz condition, which this time only
contains vector norms (and directional derivatives):
−1 F (x)
F (x̄) − F (x) (x̄ − x) ≤ ωx̄ − x2 , x, x̄ ∈ D .
(1.7)
1.2 Newton’s Method for General Nonlinear Problems
15
This assumption allows a clean affine covariant theory about the local
quadratic convergence of the Newton iterates including local uniqueness of
the solution x∗ —see Section 2.1 below. Moreover, this type of theorem will
be the stem from which a variety of computationally useful convergence theorems branch off.
Summarizing, any affine covariant convergence theorems will lead to results
in terms of iterates {xk }, correction norms Δxk or error norms xk − x∗ .
Bibliographical Note. For quite a while, affine covariance held only in
very few convergence theorems for local Newton methods, among which are
Theorem 6. (1.XVIII) in the book of Kantorovich/Akhilov [127] from 1959,
part of the theoretical results by J.E. Dennis [52, 53], or an interesting early
paper by H.B. Keller [129] from 1970 (under the weak assumption of just
Hölder continuity of F (x)). None of these authors, however, seems to have
been fully aware of the importance of this invariance property, since all of
them neglected this aspect in their later work.
A systematic approach toward affine covariance, then simply called affine
invariance, has been started in 1972 by the author in his dissertation [59],
published two years later in [60]. His initial motivation had been to overcome
severe difficulties in the actual application of Newton’s method within multiple shooting—compare Section 7.1 below. In 1979, this approach has been
transferred to convergence theory in a paper by P. Deuflhard and G. Heindl
[76]. Following the latter paper, T. Yamamoto has preserved affine covariance
in his subtle convergence estimates for Newton’s method—see, e.g., his starting paper [202] and work thereafter. Around that time H.G. Bock [29, 31, 32]
also joined the affine invariance crew and slightly improved the theoretical
characterization from [76]. The first affine covariant convergence proof for
inexact Newton methods is due to T.J. Ypma [203].
Affine contravariance. This setting is dual to the preceding one: we keep
the image space of F fixed (A = I) and consider the whole class of problems
G(y) = F (By) , x = By , B ∈ GL(n)
that is generated by the class GL(n) of nonsingular matrices B. Consequently,
a common convergence theory for the whole problem class will not lead to
statements about the Newton iterates {y k }, but only about the residuals
{F (xk )}, which are independent of any choice of B. Once more, the classical
conditions (1.2) and (1.4) can be telescoped, this time in image space terms
only:
F (x̄) − F (x) (x̄ − x) ≤ ωF (x)(x̄ − x)2 .
(1.8)
Observe that both sides are independent of B, since, for example
G (y)(ȳ − y) = F (x)B(ȳ − y) = F (x)(x̄ − x) .
16
1 Introduction
A Newton-Mysovskikh type theorem on the basis of such a Lipschitz condition
will lead to convergence results in terms of residual norms F (xk ).
Bibliographical Note. The door to affine contravariance in the Lipschitz
condition has been opened by A. Hohmann in his dissertation [120] , wherein
he exploited it for the construction of a residual based inexact Newton method
within an adaptive collocation method for ODE boundary value problems—
compare Section 7.4 below.
At first glance, the above dual affine invariance classes seem to be the only
ones that might be observed in actual computation. At second glance, however, certain couplings between the linear transformations A and B may arise,
which are discussed next.
Affine conjugacy. Assume that we have to solve the minimization problem
f (x) = min , f : D ⊂ Rn → R
for a functional f , which is convex in a neighborhood D of the minimum
point x∗ . Then this problem is equivalent to solving the nonlinear equations
F (x) = grad f (x) = f (x)T = 0 , x ∈ D .
For such a gradient mapping F the Jacobian F (x) = f (x) is symmetric
and certainly positive semi-definite. Moreover, assume that F (x) is strictly
positive definite so that F (x)1/2 can be defined. This also implies that f is
strictly convex. Upon transforming the minimization problem to
g(y) = f (By) = min , x = By ,
we arrive at the transformed equations
G(y) = B T F (By) = 0
and the transformed Jacobian
G (y) = B T F (x)B , x = By .
The Jacobian transformation is conjugate, which motivates the name of this
special affine invariance. Due to Sylvester’s theorem (compare [151]), it conserves the index of inertia, so that all G are symmetric and strictly positive
definite. Affine conjugate theoretical terms are, of course, functional values
f (x) and, in addition, so–called local energy products
(u, v) = uT F (x)v , u, v, x ∈ D .
Just note that energy products are invariant under this kind of affine transformation, since
1.2 Newton’s Method for General Nonlinear Problems
17
u, v, x → ū = Bu, v̄ = Bv, x = By
implies
uT G (y)v = ūT F (x)v̄ .
Local energy products induce local energy norms
F (x)1/2 u2 = (u, u) = uT F (x)u , u, x ∈ D .
In this framework, telescoping the theoretical assumptions (1.2) and (1.4)
leads to an affine conjugate Lipschitz condition
F (x)−1/2 F (x̄) − F (x) (x̄ − x) ≤ ωF (x)1/2 (x̄ − x)2 .
(1.9)
Affine conjugate convergence theorems will lead to results in terms of functional values f (x) and energy norms of corrections F (z)1/2 Δxk or errors
F (z)1/2 (xk − x∗ ).
Bibliographical Note. The concept of affine conjugacy dates back to
P. Deuflhard and M. Weiser, who, in 1997, defined and exploited it for the
construction of an adaptive Newton multilevel FEM for nonlinear elliptic
PDEs—see [84, 85] and Section 8.3.
Affine similarity. This invariance principle is more or less common in the
differential equation community—apart perhaps from the name given here.
Consider the case that the solution of the nonlinear system F (x) = 0 can be
interpreted as steady state or equilibrium point of the dynamical system
ẋ = F (x) .
(1.10)
Arbitrary affine transformation
Aẋ = AF (x) = 0
here affects both the domain and the image space of F in the same way—
of course, differentiability with respect to time differs. The corresponding
problem class to be studied is then
G(y) = AF (A−1 y) = 0 , y = Ax ,
which gives rise to the Jacobian transformation
G (y) = AF (x)A−1 .
This similarity transformation (which motivates the name affine similarity) is
known to leave the Jacobian eigenvalues λ invariant. Note that a theoretical
characterization of stability of the equilibrium point involves their real parts
(λ). In fact, an upper bound of these real parts, called the one-sided Lipschitz constant, will serve as a substitute of the Lipschitz constant of F , which
18
1 Introduction
is known to restrict the analysis to nonstiff differential equations. As an affine
similar representative, we may formally pick the (possibly complex) Jordan
canonical form J, known to consist of elementary Jordan blocks for each
separate eigenvalue. Let the Jacobian at any selected point x̂ be decomposed
such that
F (x̂) = T (x̂)J(x̂)T (x̂)−1 = T JT −1 ,
which implies
G (ŷ) = AF (x)A−1 = (AT )J(AT )−1 .
Consequently, any theoretical results phrased in terms of the canonical norm
| · | := T −1 · will meet the requirement of affine similarity. We must, however, remain aware
of the fact that numerical Jordan decomposition may be ill-conditioned ,
whenever eigenvalue clusters arise—a property, which is reflected in the size
of cond(T ). With this precaution, an affine similar approach will be helpful
in the analysis of stiff initial value problems for ODE’s (see Chapter 6).
In contrast to the other invariance classes, note that here not only Newton’s
iteration exhibits the correct affine similar pattern, but also any fixed point
iteration of the type
xk+1 = xk + αk F (xk ) ,
assuming the parameters αk are chosen by some affine similar criterion.
Hence, any linear combination of Newton and fixed point iteration may be
considered as well: this leads to an iteration of the type
I − τ F (xk ) (xk+1 − xk ) = τ F (xk ) ,
which is nothing else than a linearly implicit Euler discretization of the above
ordinary differential equation (1.10) with timestep τ to be adapted. As worked
out in Section 6.4, such a pseudo-transient continuation method can be safely
applied only, if the equilibrium point is dynamically stable—a condition anyway expected from geometrical insight. As a ‘first choice’, we then arrive at
the following Lipschitz condition
| (F (x̄) − F (x)) u| ≤ ω|x̄ − x||u| .
Unfortunately, the canonical norm is computationally not easily available
and at the same time may suffer from ill-conditioning—reflected in the size
of cond(T ). Therefore, upon keeping in mind that in affine similar problems
domain and image space of F have the same transformation behavior, we are
led to realize a ‘second best’ choice: we may switch from the canonical norm
| · | to the standard norm · thus obtaining a Lipschitz condition of the
structure
(F (x̄) − F (x)) u ≤ ωx̄ − x · u .
1.2 Newton’s Method for General Nonlinear Problems
19
However, in this way we lose the affine similarity property in the definition of
ω, which means we have to apply careful scaling at least. In passing, we note
that here the classical Lipschitz condition (1.4) arises directly from affine
invariance considerations; however, a bounded inverse assumption like (1.2)
is not needed in this context, but replaced by other conditions.
Scaling invariance. Scaling as discussed at the beginning of this section is
a special affine transformation. In general, we will want to realize a scaling
invariant algorithm, i.e. an algorithm that is invariant under the choice of
units in the given problem. Closer examination shows that the four different
affine invariance classes must be treated differently.
In an affine covariant setting, the formal assumption B = I will certainly
cover any fixed scaling transformation of the type B = D so that ‘dimensionless’ variables
y = D−1 x , D = diag(α1 , . . . , αn ) , αi > 0
are used at least inside the codes (internal scaling). For example, with components x = (x1 , . . . , xn ), relative scaling could mean any a-priori choice like
αi = |x0i | , if |x0i | = 0
or an iterative adaptation like
αk+1
= max |xki | , |xk+1
| .
i
i
Whenever these choices guarantee αi > 0, then scaling invariance is assured:
to see this, just re-scale the components of x according to
xi −→ x̂i = βi xi ,
which implies
αi −→ α̂i = βi αi
and leaves
yi =
x̂i
xi
=
α̂i
αi
unchanged. In reality, however, absolute threshold values αmin > 0 have to be
imposed in the form, say
ᾱi = max{αi , αmin }
to avoid overflow for values close to zero. By construction, such threshold
values spoil the nice scaling invariance property, unless they are defined for
dimensionless components of the variable y.
20
1 Introduction
In an affine contravariant setting, scaling should be applied in the image
space of F , which means for the residual components
F → G = D−1 F
with appropriately chosen diagonal matrix D.
For affine similarity, simultaneous scaling should be applied in both domain
and image space
x , F → y = D−1 x , G = D−1 F .
Finally, the affine conjugate energy products can be verified to be scaling
invariant already by construction.
Further affine invariance classes. The four affine invariance classes mentioned so far actually represent the dominant classes of interest. Beyond these,
certain combinations of these classes play a role in problems with appropriate substructures, each of which gives rise to one of the ‘grand four’. As
an example take optimization with equality constraints, which may require
affine covariance or contravariance in the constraints, but affine conjugacy
in the functional—see, e.g., the recent discussion [193] by S. Volkwein and
M. Weiser.
1.2.3 The algorithmic paradigm
The key question treated in this book is how theoretical results from convergence analysis can be exploited for the construction of adaptive Newton
algorithms. The key answer to this question is to realize affine invariant computational estimates of affine invariant Lipschitz constants that are cheaply
available in the course of the algorithms. The realization is done as follows:
We identify some theoretical local Lipschitz constant ω defined over a nonempty
domain D such that
ω = sup g(x, y, z)
(1.11)
x,y,z∈D
in terms of some scalar expression g(x, y, z) that will only contain affine
invariant terms. For ease of writing, we will mostly just write
g(x, y, z) ≤ ω for all x, y, z ∈ D ,
even though we mean the best possible estimates (1.11) to characterize nonlinearity by virtue of Lipschitz constants. Once such a g has been selected,
we exploit it by defining some corresponding computational local estimate
according to
[ω] = g(x̂, ŷ, ẑ) for specific x̂, ŷ, ẑ ∈ D .
By construction, [ω] and ω share the same affine invariance property and
satisfy the relation
[ω] ≤ ω .
1.3 A Roadmap of Newton-type Methods
21
Illustrating example. For the affine covariant Lipschitz condition (1.6) we
have
ω0 = sup g(x, y, x0 ) =
x,y∈D
F (x0 )−1 (F (x) − F (y)) .
x − y
As a local affine covariant estimate, we may choose
F (x0 )−1 F (x1 ) − F (x0 ) [ω0 ] = g(x1 , x0 , x0 ) =
x1 − x0 (1.12)
(1.13)
in terms of the anyway computed Newton iterates x0 , x1 . In actual implementation, we will apply estimates different from (1.12) and (1.13), but preferable
in the algorithmic context. The art in this kind of approach is to find out,
among many possible theoretical characterizations, those ones that give rise
to ‘cheap and suitable’ computational estimates and, in turn, lead to the
construction of efficient algorithms.
There remains some gap ω − [ω] ≥ 0, which can be reduced by appropriate
reduction of the domain D. As will turn out, efficient adaptive Newton algorithms can be constructed, if [ω] catches at least one leading binary digit
of ω—for details see the various bit counting lemmas scattered all over the
book.
Remark 1.1 If the paradigm were realized without a strict observation
of affine invariance of Lipschitz constants and estimates, then undesirable
geometrical distortion effects (like those described in detail in (1.5) ) would
lead to totally unrealistic estimates and thus could not be expected to be a
useful basis for any efficient algorithm.
Bibliographical Note. The general paradigm described here was, in an
intuitive sense, already employed by P. Deuflhard in his 1975 paper on adaptive damping for Newton’s method [63]. In 1979, the author formalized the
whole approach introducing the notation [·] for computational estimates and
exploited it for the construction of adaptive continuation methods [61]. Early
on, H.G. Bock also took up the paradigm in his work on multiple shooting techniques for parameter identification and optimal control problems
[29, 31, 32].
1.3 A Roadmap of Newton-type Methods
There is a large variety of Newton-type methods, which will be discussed in
the book and therefore named and briefly sketched here.
22
1 Introduction
Ordinary Newton method. For general nonlinear problems, the classical
ordinary Newton method reads
F (xk )Δxk = −F (xk ) , xk+1 = xk + Δxk ,
k = 0, 1, . . . .
(1.14)
For F : D ⊂ Rn → Rn a Jacobian (n, n)-matrix is required. Sufficiently accurate Jacobian approximations can be computed by symbolic differentiation
or by numerical differencing—see, for example, the automatic differentiation
due to A. Griewank [112].
The above form of the linear system deliberately reflects the actual sequence
of computation: first, compute the Newton corrections Δxk , then improve
the iterates xk to obtain xk+1 —to avoid possible cancellation of significant
digits, which might occur, if we solve for the new iterates xk+1 directly.
Simplified Newton method. This variant of Newton’s method is characterized by keeping the initial derivative throughout the whole iteration:
k
k
F (x0 )Δx = −F (xk ) , xk+1 = xk + Δx , k = 0, 1, . . . .
Compared to the ordinary Newton method, computational cost per iteration
is saved—at the possible expense of increasing the number of iterations and
possibly decreasing the convergence domain of the thus defined iteration.
Newton-like methods. This type of Newton method is characterized by
the fact that, in finite dimension, the Jacobian matrices are either replaced by
some fixed ‘close by’ Jacobian F (z) with z = x0 , or by some approximation
so that
M (xk )δxk = −F (xk ) ,
xk+1 = xk + δxk ,
k = 0, 1, . . . .
As an example, deliberate ‘sparsing’ of a large Jacobian, which means dropping of ‘weak couplings’, will permit the use of a direct sparse solver for the
Newton-like corrections and therefore possibly help to reduce the work per
iteration; if really only weak couplings are dropped, then the total iteration
pattern will not deteriorate significantly.
Exact Newton methods. Any of the finite dimensional Newton-type methods requires the numerical solution of the linear equations
F (xk )Δxk = −F (xk ) .
Whenever direct elimination methods are applicable, we speak of exact Newton methods. However, naive application of direct elimination methods may
cause serious trouble, if scaling issues are ignored.
1.3 A Roadmap of Newton-type Methods
23
Bibliographical Note. There are numerous excellent books on the numerical solution of linear systems—see, e.g., the classic by G.H. Golub and
C.F. van Loan [107]. Programs for direct elimination in full or sparse mode
can be found in the packages LAPACK [5], SPARSPAK [100], or [27]. As a rule,
these codes leave the scaling issue to the user—for good reasons, since the
user will typically know the specifications behind the problem that define the
necessary scaling.
Local versus global Newton methods. Local Newton methods require
‘sufficiently good’ initial guesses. Global Newton methods are able to compensate for bad initial guesses by virtue of damping or adaptive trust region
strategies. Exact global Newton codes for the solution of nonlinear equations
are named NLEQ plus a characterizing suffix. We give details about
• NLEQ-RES for the residual based approach,
• NLEQ-ERR for the error oriented approach, or
• NLEQ-OPT for convex optimization.
Inexact Newton methods. For extremely large scale nonlinear problems
the arising linear systems for the Newton corrections can no longer be solved
directly (‘exactly’), but must be solved iteratively (‘inexactly’)—which gives
the name inexact Newton methods. The whole scheme then consists of an
inner iteration (at Newton step k)
F (xk ) δxki = −F (xk ) + rik ,
k = 0, 1, . . . ,
xk+1
= xk + δxki ,
i
i = 0, 1, . . . , ikmax
(1.15)
in terms of residuals rik and an outer iteration where, given x0 , the iterates
are defined as
xk+1 = xk+1
i
for i = ikmax , k = 0, 1, . . . .
Compared with the exact Newton corrections in (1.14), errors δxki − Δxk
arise. Throughout the book, we will mostly drop the inner iteration index i
for ease of notation.
In an adaptive inexact Newton method, the accuracy of the inner iteration
should be matched to the outer iteration, preferably such that the Newton
convergence pattern is essentially unperturbed—which means an appropriate
control of imax above. Criteria for the choice of the truncation index imax
depend on affine invariance, as will be worked out in detail. With this aspect
in mind, inexact Newton methods are sometimes also called truncated Newton
methods.
Inexact global Newton codes for the solution of large scale nonlinear equations
are named GIANT plus a suffix characterizing the combination with an inner
iterative solver. The name GIANT stands for Global Inexact Affine invariant
Newton Techniques. We will work out details for
24
1 Introduction
• GIANT-GMRES for the residual based approach,
• GIANT-CGNE and GIANT-GBIT for the error oriented approach, or
• GIANT-PCG for convex optimization.
As for the applied iterative solvers, see Section 1.4 below.
Preconditioning. A compromise between direct and iterative solution of
the arising linear Newton correction equations is obtained by direct elimination of ‘similar’ linear systems, which can be used in a wider sense than just
scaling as mentioned above. For its characterization we write
−1
CL F (xk )CR CR
δxki = −CL F (xk ) − rik ,
i = 0, 1, . . . imax (1.16)
or, equivalently, also
k
−1
CL F (xk )CR CR
δxi − Δxki = CL rik ,
i = 0, 1, . . . imax .
Consequently, within the algorithms any residual or error norms need to be
replaced by their preconditioned counterparts
k
−1
rik , δxki − Δxki −→ CL rik , CR
δxi − Δxki .
Matrix-free Newton methods. Linear iterative solvers within inexact
Newton methods only require the evaluation of Jacobian matrix vector products so that numerical difference approximations
. F (x + δv) − F (x)
F (x)v =
δ
can be conveniently realized. Note, however, that the quality of such directional difference approximations will heavily depend on the choice of the
relative deviation parameter δ and the mantissa length of the used arithmetic. A numerically stable realization will use automatic differentiation as
suggested by A. Griewank [112].
Secant method. For scalar equations, say f (x) = 0, this type of method is
derived from Newton’s method by substituting the tangent by the secant
f (xk + δxk ) −→
f (xk + δxk ) − f (xk )
= jk+1
δxk
and computing the correction as
δxk+1 = −
f (xk+1 )
, xk+1 = xk + δxk .
jk+1
The thus constructed secant method is known to converge locally superlinearly.
1.3 A Roadmap of Newton-type Methods
25
Quasi-Newton methods. This class of methods extends the secant idea
to systems of equations. In this case only a so-called secant condition
Jδxk = F (xk+1 ) − F (xk )
(1.17)
can be imposed, wherein J represents some Jacobian approximation to be
specified. The above condition does not determine a unique J, but a whole
class of matrices. If we recur to the previous quasi-Newton step as
Jk δxk = −F (xk ) ,
we may select special Jacobian rank-1 updates as
Jk+1 = Jk +
F (xk+1 )z T
,
z T δxk
z ∈ Rn , z = 0 ,
where the vector z is arbitrary, in principle. As will be shown below in detail,
the specification of z is intimately linked with affine invariance. Once z has
been specified, the next quasi-Newton step
Jk+1 δxk+1 = −F (xk+1 )
is determined. In the best case, superlinear local convergence can be shown to
hold again. A specification to linear systems is the algorithm GBIT described
in Section 1.4.4 below.
Gauss-Newton methods. This type of method applies to nonlinear least
squares problems, whether unconstrained or constrained. The method requires the nonlinear least squares problems to be statistically well-posed,
characterized either as ‘small residual’ (Section 4.2.1) or as ‘adequate’ problems (Section 4.3.2). For this problem class, local Gauss-Newton methods
are appropriate, when ‘sufficiently good’ initial guesses are at hand, while
global Gauss-Newton methods are used, when only ‘bad initial guesses’ are
available. In the statistics community Gauss-Newton methods are also called
scoring methods.
Quasilinearization. Infinite dimensional Newton methods for operator
equations are also called Newton methods in function space or quasilinearization. The latter name stems from the fact that the nonlinear operator equation is solved via a sequence of corresponding linearized operator equations.
Of course, the linearized equations for the Newton corrections can only be
solved approximately. Consequently, inexact Newton methods supply the correct theoretical frame, within which now the ‘truncation errors’ represent
approximation errors, typically discretization errors.
26
1 Introduction
Inexact Newton multilevel methods. We reserve this term for those multilevel schemes, wherein the arising infinite dimensional linear Newton systems are approximately solved by some linear multilevel or multigrid method;
in such a setting, Newton methods act in function space. The highest degree of
sophistication of an inexact Newton multilevel method would be an adaptive
Newton multilevel method, where the approximation errors are controlled
within an abstract framework of inexact Newton methods.
Multilevel Newton methods. Unfortunately, the literature is often not
unambiguous in the choice of names. In particular, the name ‘Newton multigrid method’ is often given to schemes, wherein a finite dimensional Newton
multigrid method is applied on each level—see, e.g., the classical textbook
[113] by W. Hackbusch or the more recent treatment [135] by R. Kornhuber,
who uses advanced functional analytic tools. In order to avoid confusion, such
a scheme will here be named ‘multilevel Newton method’.
Nonlinear multigrid methods. For the sake of clarity, it may be worth
mentioning that ‘nonlinear multigrid methods’ are not Newton methods, but
fixed point iteration methods, and therefore not treated within the scope of
this book.
Bibliographical Note. The classic among the textbooks for the numerical solution of finite dimensional systems of nonlinear equations has been
the 1970 book of J.M. Ortega and W.C. Rheinboldt [163]. It has certainly
set the state of the art for quite a long time. The monograph [177] by
W.C. Rheinboldt guides into related more recent research areas. The popular textbook [132] by C.T. Kelley offers a nice introduction into finite dimensional inexact Newton methods—see also references therein. The technique of
‘preconditioning’ is usually attributed to O. Axelsson—see his textbook [11]
and references therein. Multigrid Newton methods are worked out in detail
in the meanwhile classic text of W. Hackbusch [113]; a detailed convergence
analysis of such methods for certain smooth as well as a class of non-smooth
problems has been recently given by R. Kornhuber [135].
1.4 Adaptive Inner Solvers for Inexact Newton Methods
As stated in Section 1.3 above, inexact Newton methods require the linear
systems for the Newton corrections to be solved iteratively. Different affine
invariance concepts naturally go with different concepts for the iterative solution. In particular, recall that
• residual norms go with affine contravariance,
• error norms go with affine covariance,
1.4 Adaptive Inner Solvers for Inexact Newton Methods
27
• energy norms go with affine conjugacy.
For the purpose of this section, let the inexact Newton system (1.15) be
written as
Ayi = b − ri ,
i = 0, 1, . . . imax
in terms of iterative approximations yi for the solution y and iterative residuals ri . In order to control the number imax of iterations, several termination
criteria may be realized:
• Terminate the iteration as soon as the residual norm ri is small enough.
• Terminate the iteration as soon as the iterative error norm y −yi is small
enough.
• If the matrix A is symmetric positive definite, terminate the iteration as
soon as the energy norm A1/2 (y − yi ) of the error is small enough.
In what follows, we briefly sketch some of the classical iterative linear solvers
with particular emphasis on appropriate termination criteria for use within
inexact Newton algorithms. We will restrict our attention to those iterative
solvers, which minimize or, at least, reduce
• the residual norm (GMRES, Section 1.4.1),
• the energy norm of the error (PCG, Section 1.4.2), and
• the error norm (CGNE, Section 1.4.3, and GBIT, Section 1.4.4).
We include the less known solver GBIT, since it is a quasi-Newton method
specialized to the solution of linear systems.
Preconditioning. This related issue deals with the iterative solution of
systems of the kind
−1
CL ACR CR
yi = CL (b − ri ),
i = 0, 1, . . . imax ,
(1.18)
where left preconditioner CL and right preconditioner CR arise. A proper
choice of preconditioner will exploit information from the problem class under
consideration and often crucially affect the convergence speed of the iterative
solver.
Bi-CGSTAB. Beyond the iterative algorithms selected here, there are numerous further ones of undoubted merits. An example is the iterative solver
Bi-CG and its stabilized variant Bi-CGSTAB due to H.A. van der Vorst [189].
This solver might actually be related to affine similarity as treated above
in Section 1.2; as a consequence, this code would be a natural candidate
within an inexact pseudo–continuation method (see Section 6.4.2). However,
this combination of inner and outer iteration would require a rather inconvenient norm (Jordan canonical norm). That is why we do not incorporate this
candidate here. However, further work along this line might be promising.
28
1 Introduction
Bibliographical Note. A good survey on many aspects of the iterative
solution of linear equation systems can be found in the textbook [181] by
Y. Saad. Preconditioning techniques are described, e.g., in the textbook [11]
by O. Axelsson.
Multilevel discretization. For the adaptive realization of inexact Newton
methods in function space, discretizations on successively finer levels play the
role of the inner iteration. That is why we additionally treat linear multigrid
methods in Section 1.4.5 below. Skipping any technical details here, multilevel methods permit an adaptive control of discretization errors on each
level—for example, see Section 7.3.3 on Fourier-Galerkin methods for periodic
orbit computation, Section 7.4.2 on polynomial collocation methods for ODE
boundary value problems, and Section 8.3 on adaptive multigrid methods for
elliptic PDEs.
1.4.1 Residual norm minimization: GMRES
A class of iterative methods aims at the successive reduction of the residual norms ri for increasing index i. Outstanding candidates among these
are those solvers that even minimize the residual norms over some Krylov
subspace—such as GMRES and CGNR. Since algorithm GMRES requires less matrix/vector multiplies per step, we focus our attention on it here.
Algorithm GMRES. Given an initial approximation y0 ≈ y compute the
initial residual r0 = b − Ay0 . Set β = r0 2 , v1 = r0 /β , V1 = v1 .
For i = 1, 2, . . . , imax :
I. Orthogononalization:
v̂i+1 = Avi − Vi hi
where hi = ViT Avi
II. Normalization:
vi+1 = v̂i+1 /v̂i+1 2
III. Update:
Vi+1 = (Vi vi+1 )
Hi =
Hi−1
0
hi
v̂i+1 2
Hi is an (i + 1, i)-Hessenberg matrix
(for i = 1 drop the left block column)
IV. Least squares problem for zi :
βe1 − Hi z = min
V. Approximate solution (i = imax ):
yi = Vi zi + y0
Array storage. Up to iteration step i, the above implementation requires
to store i + 2 vectors of length n.
1.4 Adaptive Inner Solvers for Inexact Newton Methods
29
Computational amount. In each iteration step i, there is one matrix/vector
multiply needed. Up to step i, the Euclidean inner products sum up to ∼ i2 n
flops.
As already stated, this algorithm minimizes the residual norms over the
Krylov subspace
Ki (r0 , A) = span {r0 , . . . , Ai−1 r0 } .
By construction, the inner residuals will decrease monotonically
ri+1 2 ≤ ri 2 .
Therefore, a reasonable inner termination criterion will check whether the
final residual ri 2 is ‘small enough’. Moreover, starting with arbitrary initial
guess y0 and initial residual r0 = 0, we have the orthogonality relation (in
terms of the Euclidean inner product · , · )
ri , ri − r0 = 0 ,
which directly implies that
r0 22 = r0 − ri 22 + ri 22
(1.19)
throughout the inner iteration. If we define
ηi =
ri 2
,
r0 2
then we will generically have ηi < 1 for i > 0 and
ηi+1 < ηi ,
if ηi = 0 .
This implies that, after a number of iterations, any adaptive truncation criterion
ηi ≤ η̄
for a prescribed threshold value η̄ < 1 can be met. In passing we note that
then (1.19) can be rewritten as
r0 − ri 22 = (1 − ηi2 )r0 22 .
(1.20)
These detailed results are applied in Sections 2.2.4 and 3.2.3.
Preconditioning. Finally, if (1.18) is applied, then the Euclidean norms of
the preconditioned residuals r̄k = CL rk are iteratively minimized in GMRES,
whereas CR only affects the rate of convergence. Therefore, if strict residual
minimization is aimed at, then only right preconditioning should be implemented, which means CL = I.
30
1 Introduction
Bibliographical Note. This iterative method has been designed as
a rather popular code by Y. Saad and M.H. Schultz [182] in 1986; an
earlier often overlooked derivation has been given by G.I. Marchuk and
Y.A. Kuznetsov [146] already in 1968.
1.4.2 Energy norm minimization: PCG
With A symmetric positive definite, we are able to define the energy product
(·, ·) and its induced energy norm · A by
(u, v) = u, Av,
u2A = (u, u)
in terms of the Euclidean inner product ·, ·. Let B ≈ A−1 denote some preconditioning matrix, assumed to be also symmetric positive definite. Usually
the numerical realization of z = Bc is much simpler and faster than the soT
lution of Ay = b. Formally speaking, we may specify some CL = CR
= B 1/2 .
This specification does not affect the energy norms, but definitely the speed
of convergence of the iteration. Any preconditioned conjugate gradient (PCG)
method reads:
Algorithm PCG. For given approximation y0 ≈ y compute the initial residual
r0 = b − Ay0 and the preconditioned residual r̄0 = Br0 . Set p0 = r̄0 σ0 =
r0 , r̄0 = r0 2B .
For i = 0, 1, . . . , imax :
αi
=
yi+1
=
γi2
=
ri+1
=
σi+1
=
pi+1
=
pi 2A
σi
1
y i + pi
αi
σi
(energy error contribution yi+1 − yi 2A )
αi
1
ri − Api , r̄i+1 = Bri+1
αi
σi+1
ri+1 2B ,
βi+1 =
σi
r̄i+1 + βi+1 pi .
Array storage. Up to iteration step i, ignoring any preconditioners, the
above implementation requires to store only 4 vectors of length n.
Computational amount. In each iteration step i, there is one matrix/vector
multiply needed. Up to step i, the Euclidean inner products sum up to ∼ 5in
flops.
This iteration successively minimizes the energy error norm y − yiA within
the associated Krylov subspace
1.4 Adaptive Inner Solvers for Inexact Newton Methods
31
Ki (r0 , A) = span{r0 , . . . , Ai−1 r0 } .
By construction, we have the orthogonality relations (also called Galerkin
conditions)
(yi − y0 , yi+m − yi )A = 0 , m = 1, . . . ,
which imply the orthogonal decompositions (with m = 1)
yi+1 − y0 2A = yi+1 − yi 2A + yi − y0 2A
(1.21)
and (with m = n − i and yn = y)
y − y0 2A = y − yi 2A + yi − y0 2A .
(1.22)
From (1.21) we easily derive that
i−1
yi − y0 2A =
i−1
yj+1 − yj 2A =
j=0
γj2 .
(1.23)
j=0
Together with (1.22), we then obtain
n−1
i = y − yi 2A =
γj2 .
(1.24)
j=i
Estimation of PCG error. Any adaptive affine conjugate inexact Newton
algorithm will require a reasonable estimate for the errors i to be able to
exploit the theoretical convergence results. Note that the monotonicity
yi − y0 A ≤ yi+1 − y0 A ≤ · · · ≤ y − y0 A
can be derived from (1.21)—a saturation effect easily observable in actual
computation. There are two basic methods to estimate the error.
(I) Assume we have a computable upper bound
0 ≥ y − y0 2A ,
¯
such as the ones suggested by G.H. Golub and G. Meurant [104] or by
B. Fischer [92]. Then, with (1.22) and (1.23), we obtain the computable upper
bound
i−1
i ≤ ¯0 −
γj2 = [i ] .
j=0
(II) As an alternative (see [68]), we may exploit the structure of (1.24) via
the lower bound
i+m
γj2 ≤ i
[i ] =
j=i
(1.25)
32
1 Introduction
for some sufficiently large index m > 0—which means to continue the iteration further just for the purpose of getting some error estimate. In the case
of ‘fast’ convergence (usually for ‘good’ preconditioners only), few terms in
the sum will suffice. Typically, we use this technique, since it does not require
any choice of an upper bound ¯0 .
Both techniques inherit the monotonicity [i+1 ] ≤ [i ] from i+1 ≤ i . Hence,
generically, after a number of iterations, any adaptive truncation criterion
[i ] ≤ for a prescribed threshold value can be met. In the inexact Newton-PCG
algorithms to be worked out below we will use the relative energy error norms
defined, for i > 0, as
[i ]
y − yi A
δi =
≈
.
yi A
yi A
Whenever y0 = 0, then (1.21) implies the monotone increase yi+1 A ≥ yi A
and therefore the monotone decrease δi+1 ≤ δi . This guarantees that any
relative error criterion
δi ≤ δ
can be met. Moreover, in this case we have the relation
yi2A = (1 + δi2 )y2A .
(1.26)
For y0 = 0, the above monotonicities and (1.26) no longer hold. This option,
however, is not used in the inexact Newton-PCG algorithms to be derived in
Sections 2.3.3 and 3.4.3 below.
1.4.3 Error norm minimization: CGNE
Another class of iterative solvers aims at the successive reduction of the
(possibly scaled) Euclidean error norms y − yi for i = 0, 1, . . .. Among these
the ones that minimize the error norms over some Krylov subspace play a
special role. For nonsymmetric Jacobian matrices the outstanding candidate
with this feature seems to be CGNE. For its economic implementation, we
recommend Craig’s variant (see, e.g., [181]), which reads:
Algorithm CGNE. Given an initial approximation y0 , compute the initial
residual r0 = b − Ay0 and set p0 = 0, β0 = 0, σ0 = r0 2 .
For i = 1, 2, . . . , imax :
1.4 Adaptive Inner Solvers for Inexact Newton Methods
pi
=
AT ri−1 + βi−1 pi−1
αi
=
σi−1 /pi2
2
γi−1
=
αi σi−1
yi
=
yi−1 + αi pi
ri
=
ri−1 − αi Api
σi
=
ri 2
βi
=
σi /σi−1
33
(Euclidean error contribution yi − yi−1 2 )
Array storage. Up to iteration step i, the above implementation requires
to store only 3 vectors of length n.
Computational amount. In each iteration step i, there are two matrix/vector multiplies needed. Up to step i, the Euclidean inner products
sum up to ∼ 5in flops.
This iteration successively minimizes the Euclidean norms y − yi within
the Krylov subspace
Ki (AT r0 , AT A) = span{AT r0 , . . . , (AT A)i−1 AT r0 } .
By construction, we have the orthogonality relations (also: Galerkin conditions)
(yi − y0 , yi+m − yi ) = 0 , m = 1, . . . ,
which imply the orthogonal decomposition (with m = 1)
yi+1 − y0 2 = yi+1 − yi 2 + yi − y0 2
(1.27)
and (with m = n − i and yn = y)
y − y0 2 = y − yi 2 + yi − y0 2 .
(1.28)
From (1.27) we easily derive that
i−1
yi − y0 2 =
i−1
yj+1 − yj 2 =
j=0
γj2 .
(1.29)
j=0
Together with (1.28), we then obtain
n−1
i = y − yi 2 =
γj2 .
j=i
(1.30)
34
1 Introduction
Estimation of CGNE error. Any adaptive affine covariant inexact Newton
algorithm will require a reasonable estimate for the errors i to be able to
exploit the theoretical convergence results. Again, the saturation effect from
the monotonicity
yi − y0 ≤ yi+1 − y0 ≤ · · · ≤ y − y0 can be derived from (1.27).
There are two basic methods to estimate the error within CGNE.
(I) Assume we have a computable upper bound
¯0 ≥ y − y0 2
in the spirit of those suggested by G.H. Golub and G. Meurant [104] or by
B. Fischer [92]. From this, with (1.28) and (1.29), we obtain the computable
upper bound
i−1
i ≤ ¯0 −
γj2 = [i ] .
j=0
(II) As an alternative, transferring an idea from [68], we may exploit the
structure of (1.30) to look at the lower bound
i+m
γj2 ≤ i
[i ] =
(1.31)
j=i
for some sufficiently large index m > 0—which means to continue the iteration further just for the purpose of getting some error estimate. In the case
of ‘fast’ convergence (for ‘sufficiently good’ preconditioner, see below), only
few terms in the sum will be needed. Typically, we use this second technique.
Both techniques inherit the monotonicity [i+1 ] ≤ [i ] from i+1 ≤ i . After
a number of iterations, any adaptive truncation criterion
[i ] ≤ for a prescribed threshold value can generically be met. In the inexact
Newton-ERR algorithms to be worked out below we will use the relative
error norms defined, for i > 0, as
δi =
[i ]
y − yi ≈
.
yi yi (1.32)
Whenever y0 = 0, then (1.27) implies the monotonicities yi+1 ≥ yi and
δi+1 ≤ δi . The latter one guarantees that the relative error criterion
δi ≤ δ
(1.33)
1.4 Adaptive Inner Solvers for Inexact Newton Methods
35
can be eventually met. For this initial value we also have the relation
yi 2 = (1 + δi2 )y2 .
(1.34)
These detailed results enter into the presentation of local inexact NewtonERR methods in Section 2.1.5.
For y0 = 0, the above monotonicities as well as the relation (1.34) no longer
hold. This situation occurs in the global inexact Newton-ERR method to be
derived in Section 3.3.4. Since the yi eventually approach y, we nevertheless require the relative truncation criterion (1.33).
Preconditioning. Finally, if (1.18) is applied, then the norms of the iterative
−1
preconditioned errors CR
(y − yi ) are minimized. Therefore, if strict unscaled
error minimization is aimed at, then only left preconditioning should be realized. In addition, if ‘good’ preconditioners CR or CL are available (resulting
in ‘fast’ convergence), then the simplification
−1
−1
CR
(y − yi ) ≈ CR
(yi+1 − yi )
will be sufficient.
Remark 1.2 Numerical experiments with large discretized PDEs in Section
8.2.1 document a poor behavior of CGNE, which seems to stem from a rather
sensitive dependence on the choice of preconditioner. Generally speaking, a
preconditioner, say B, is expected to reduce the condition number κ(J) to
some κ(BJ) κ(J). However, as the algorithm CGNE works on the normal
equations, the characterizing dependence is on κ2 (BJ) κ(BJ). In contrast
to this behavior, the preconditioned GMRES just depends on κ(BJ) as the
characterizing quantity.
1.4.4 Error norm reduction: GBIT
The quasi-Newton methods already mentioned in Section 1.3 can be specified to apply to linear systems as well. Following the original paper by
P. Deuflhard, R. Freund, and A. Walter [74], a special affine covariant rank-1
update can be chosen, which turns out to be Broyden’s ‘good’ update [40].
Especially for linear systems, an optimal line search is possible, which then
gives the algorithm GBIT (abbreviation for Good Broyden ITerative solver for
linear systems).
Preconditioner improvement. The main idea behind this algorithm is to
improve any (given) initial preconditioner B0 ∼ A, or H0 ∼ A−1 , respectively, successively to Bi ∼ A, Hi ∼ A−1 . Let Ei = I − A−1 Bi denote the
preconditioning error, then each iterative step can be shown to realize some
new preconditioner such that
36
1 Introduction
Ei+1 2 ≤ Ei 2 ,
i = 0, 1, . . . .
Error reduction. In [74], this algorithm has been proven to converge under
the sufficient assumption
E0 2 < 13 ,
(1.35)
in the sense that
yi+1 − y < yi − y .
(1.36)
Moreover, asymptotic superlinear convergence can even be shown. Numerical
experience shows that an assumption weaker than (1.35) might do, but there
is no theoretical justification of such a statement yet.
The actual implementation of the algorithm does not store the improved
preconditioners Hi explicitly, but exploits the Sherman-Morrison formula to
obtain a cheap recursion. The following implementation is a recent slight
improvement over the algorithm GB suggested in [74]—essentially replacing
the iterative stepsize ti = τi therein by some modification (see below and
Exercise 1.4 for τmax = 1).
Algorithm GBIT. Given an initial guess y0 , an initial preconditioner H0 ∼
A−1 , and some inner product u, v.
Initialization:
r0
=
b − Ay0
Δ0
=
H0 r0
σ0
=
Δ0 , Δ 0 Iteration loop i = 0, 1, . . . , imax :
qi
= AΔi
ζ0
= H0 qi
Update loop m = 0, . . . , i − 1 (for i ≥ 1):
ζm+1 = ζm +
if
zi
=
γi
=
τi
=
σi /γi
τi
<
τmin :
ti
=
τi
Δm , ζm (Δm+1 − (1 − tm )Δm )
σm
ζi
Δi , z i restart
1.4 Adaptive Inner Solvers for Inexact Newton Methods
if
if
ti
> τmax :
yi+1
= y i + t i Δi
(ri+1
= ri − ti qi )
Δi+1
= (1 − ti + τi )Δi − τi zi
σi+1
=
i
=
i
≤ ρyi+1 · ERRTOL :
37
ti = 1
Δi+1 , Δi+1 1√
2 σi−1
+ 2σi + σi+1
solution found
The parameters τmin , τmax are set internally such that 0 < τmin 1, τmax ≥ 1,
the safety factor ρ < 1 and the error tolerance ERRTOL are user prescribed
values.
Array storage. Up to iteration step i, the above recursive implementation
requires to store the i + 3 vectors
Δ0 , . . . , Δi , q, z ≡ ζ .
of length n.
Computational amount. In each iteration step i, the computational work
is dominated by one matrix/vector multiply, one solution of a preconditioned
system (ζ0 = H0 q), and ∼ 2i · n flops for the Euclidean inner product. Up to
step i, this sums up to i preconditioned systems and ∼ i2 n flops.
Inner product. Apart from the Euclidean inner product u, v = uT v any
scaled version such as u, v = (D−1 u)T D−1 v, with D a diagonal scaling matrix, will be applicable—and even preferable. For special problems like discrete PDE boundary value problems certain discrete L2 -products and norms
are recommended.
Error estimation and termination criterion. By construction, we obtain
the relation
yi − y = Δi − Ei Δi ,
which, under the above preconditioning assumption, certainly implies the
estimation property
Ei Δi Ei Δi 1−
Δi ≤ yi − y ≤ 1 +
Δi .
Δi Δi Hence, the true error can be roughly estimated as
√
yi − y ≈ Δi = σi .
In order to suppress possible outliers, an average of the kind
i =
1
2
σi−1 + 2σi + σi+1
(1.37)
38
1 Introduction
is typically applied for i > 1. This estimator leads us to the relative termination criterion
i ≤ ρyi+1 · ERRTOL ,
as stated above.
Note that the above error estimator cannot be shown to inherit the monotonicity property (1.36). Consequently, this algorithm seems to be less efficient than CGNE within the frame of Newton-ERR algorithms. Surprisingly,
this expectation is not at all in agreement with numerical experiments—see,
for instance, Section 8.2.1.
Remark 1.3 On top of GBIT we also applied ideas of D.M. Gay and
R.B. Schnabel [97] about successive orthogonalization of update vectors to
construct some projected ‘good’ Broyden method for linear systems. The
corresponding algorithm PGBIT turned out to require only slightly less iterations, but significantly more array storage and computational amount—and
is therefore omitted here.
1.4.5 Linear multigrid methods
In Newton methods for operator equations the corresponding iterates and solutions live in appropriate infinite dimensional function spaces. For example,
in steady state partial differential equations (PDEs), the solutions live in some
Sobolev space—like H α —depending on the prescribed boundary conditions.
It is an important mathematical paradigm that any such infinite dimensional
space should not just be represented by a single finite dimensional space of
possibly high dimension, but by a sequence of finite dimensional subspaces
with increasing dimension.
Consequently, any infinite dimensional Newton method will be realized via a
sequence of finite dimensional linearized systems
Ayj = b + rj ,
j = 0, 1, . . . jmax ,
where the residuals rj represent approximation errors, mostly discretization
errors. Each of the subsystems is again solved iteratively, which gives rise to
the question of accuracy matching of discretization versus iteration. This is
the regime of linear multigrid or multilevel methods—see, e.g., the textbook
of W. Hackbusch [113] and Chapter 8 below on Newton multilevel methods.
Adaptivity. In quite a number of application problems rather localized
phenomena occur. In this case, uniform grids are by no means optimal, which,
in turn, also means that the classical multigrid methods on uniform grids
could not be regarded as optimal. For this reason, multigrid methods on
adaptive grids have been developed quite early, probably first by R.E. Bank
1.4 Adaptive Inner Solvers for Inexact Newton Methods
39
[18] in his code PLTMG and later in the family UG of parallel codes [22, 21] by
G. Wittum, P. Bastian, and their groups.
Independent of the classical multigrid methods, a multilevel method based
on conjugate gradient iteration with some hierarchical basis (HB) preconditioning had been suggested for elliptic PDEs by H. Yserentant [204]. An
adaptive 2D version of the new method had been first designed and implemented by P. Deuflhard, P. Leinen, and H. Yserentant [78] in the code
KASKADE. A more mature version including also 3D has been worked out by
F. Bornemann, B. Erdmann, and R. Kornhuber [36]. The present version of
KASKADE [23] contains the original HB-preconditioner for 2D and the more
recent BPX-preconditioner due to J. Xu [200, 39] for 3D.
Additive versus multiplicative multigrid methods. In the interpretation of multigrid methods as abstract Schwarz methods as given by J. Xu
[201], which the author prefers to adopt, the classical multigrid methods
are now called multiplicative multigrid methods, whereas the HB- or BPXpreconditioned conjugate gradient methods are called additive multigrid
methods. In general, any difference in speed between additive or multiplicative multigrid methods is only marginal, since the bulk of computing time is
anyway spent in the evaluation of the stiffness matrix elements and the right
hand side elements. For the orientation of the reader: UG is nearly exclusively
multiplicative, PLTMG is predominantly multiplicative with some additive options, KASKADE is predominantly additive with some multiplicative code for
special PDE eigenvalue problems.
Cascadic multigrid methods. These rather recent multigrid methods can
be understood as a confluence of additive and multiplicative multigrid methods. From the additive point of view, cascadic multigrid methods are characterized by the simplest possible preconditioner: either no or just a diagonal
preconditioner is applied; as a distinguishing feature, coarser levels are visited more often than finer levels—to serve as preconditioning substitutes.
From the multiplicative side, cascadic multigrid methods may be understood
as multigrid methods with an increased number of smoothing iterations on
coarser levels, but without any coarse grid corrections. A first algorithm of
this type, the cascadic conjugate gradient method (algorithm CCG) had been
proposed by the author in [68]. First rather restrictive convergence results
were due to V. Shaidurov [185]. The general cascadic multigrid method class
with arbitrary inner iterations beyond conjugate gradient methods has been
presented by F. Bornemann and P. Deuflhard [35].
Just to avoid mixing terms: cascadic multigrid methods are different from
the code KASKADE, which predominantly realizes additive multigrid methods.
Local error estimators. Any efficient implementation of adaptive multigrid methods (additive, multiplicative, cascadic) must be based on cheap local
40
1 Introduction
error estimators or, at least, local error indicators. In the best case, these are
derived from theoretical a-posteriori error estimates. These estimates will be
local only, if local (right hand side) perturbations in the given problem remain
local—i.e., if the Greens’ function of the PDE problem exhibits local behavior.
As a consequence of this elementary insight, adaptive multigrid methods will
be essentially applicable to linear or nonlinear elliptic problems (among the
stationary PDE problems). A comparative assessment of the different available local error estimators has been given by F. Bornemann, B. Erdmann,
and R. Kornhuber in [37]. In connection with any error estimator, the local extrapolation method due to I. Babuška and W.C. Rheinboldt [14] can
be applied. The art of refinement is quite established in 2D (see the ‘red’
and ‘green’ refinements due to R.E. Bank et al. [20]) and still under further
improvement in 3D.
Summarizing, adaptive multilevel methods for linear PDEs play a dominant role in the frame of adaptive Newton multilevel methods for nonlinear
PDEs—see, e.g., Section 8.3.
Exercises
Exercise 1.1 Given a nonlinear C 1 -mapping F : X → Y over some domain
D ⊂ X for Banach spaces X, Y , each endowed with some norm · . Assume
a Lipschitz condition of the form
F (x) − F (y) ≤ γx − y , x, y ∈ D .
Let the derivative at some point x0 have a bounded inverse with
F (x0 )−1 ≤ β0 .
Show that then, for all arguments x ∈ D in some open ball S(x0 , ρ) with
1
ρ=
, there exists a bounded derivative inverse with
β0 γ
F (x)−1 ≤
β0
.
1 − β0 γx − x0 Exercise 1.2 Usual proofs of the implicit function theorem apply the
Newton-Kantorovich theorem—compare Section 1.2. Revisit this kind of
proof in any available textbook in view of affine covariance. In particular,
replace condition (1.3) for a locally bounded inverse and Lipschitz condition
(1.4) by some affine covariant Lipschitz condition like (1.6), which defines
Exercises
41
some local affine covariant Lipschitz constant ω0 . Formulate the thus obtained affine covariant implicit function theorem. Characterize the class of
problems, for which ω0 = ∞.
Exercise 1.3
Consider the scalar monomial equation
f (x) = xm − a = 0 .
We want to study the convergence properties of Newton’s method. For this
purpose consider the general corresponding contraction term
Θ(x) =
f f .
f 2
Verify this expression in general and calculate it for the specific case. What
kind of convergence occurs for m = 1? How could the Newton method be ‘repaired’ such that quadratic convergence still occurs? Why is this, in general,
not a good idea?
Hint: Study the convergence properties under small perturbations.
Exercise 1.4 Consider the linear iterative solver GBIT described in Section
1.4.4. In the notation introduced there, let the iterative error be written as
ei = y − yi .
a) Verify the recursive relation
ei+1 = (1 − ti )ei + ti E i ei ,
where
E i = −(I − Ei )−1 Ei .
b) Show that, under the assumption E i < 1 on a ‘sufficiently good’ preconditioner, any stepsize choice 0 < ti ≤ 1 will lead to convergence, i.e.,
ei+1 < ei ,
if ei = 0 .
c) Verify that Ei+1 2 ≤ Ei 2 holds, so that E0 2 < 12 implies Ēi 2 < 1
for all indices i = 0, 1, . . . .
d) Compare the algorithm GBIT for the two steplength strategies ti = τi
and ti = min(1, τi ) at your favorite linear system with nonsymmetric
matrix A.
Part I
ALGEBRAIC EQUATIONS
2 Systems of Equations: Local Newton
Methods
This chapter deals with the numerical solution of systems of nonlinear equations with finite, possibly large dimension n. The term local Newton methods refers to the situation that—only throughout this chapter—‘sufficiently
good’ initial guesses of the solution are assumed to be at hand. Special attention is paid to the issue of how to recognize—in a computationally cheap
way—whether a given initial guess x0 is ‘sufficiently good’. As it turns out,
different affine invariant Lipschitz conditions, which have been introduced in
Section 1.2.2, lead to different characterizations of local convergence domains
in terms of error oriented norms, residual norms, or energy norms and convex functionals, which, in turn, give rise to corresponding variants of Newton
algorithms.
We give three different, strictly affine invariant convergence analyses for the
cases of affine covariant (error oriented) Newton methods (Section 2.1), affine
contravariant (residual based) Newton methods (Section 2.2), and affine conjugate Newton methods for convex optimization (Section 2.3). Details are
worked out for ordinary Newton algorithms, simplified Newton algorithms,
and inexact Newton algorithms—synoptically for the three affine invariance
classes. Moreover, affine covariance appears as associated with Broyden’s
‘good’ quasi-Newton method, whereas affine contravariance corresponds to
Broyden’s ‘bad’ quasi-Newton method.
2.1 Error Oriented Algorithms
A convergence analysis for any error oriented algorithm of Newton type will
start from affine covariant Lipschitz conditions of the kind (1.7) and lead to
results in the space of the iterates only. The behavior of the residuals will
be ignored. For actual computation, scaling of any arising norms of Newton
corrections is tacitly assumed.
2.1.1 Ordinary Newton method
Consider the ordinary Newton method in the notation
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance
and Adaptive Algorithms, Springer Series in Computational Mathematics 35,
DOI 10.1007/978-3-642-23899-4_2, © Springer-Verlag Berlin Heidelberg 2011
45
46
2 Systems of Equations: Local Newton Methods
F (xk )Δxk = −F (xk ) , xk+1 = xk + Δxk ,
k = 0, 1, . . . .
(2.1)
Convergence analysis. Because of its fundamental importance, we begin
with an affine covariant version of the classical ‘Newton-Kantorovich theorem’. Only at this early stage we state the theorem in Banach spaces—well
aware of the fact that a Banach space formulation is not directly applicable to
numerical methods: in the numerical solution of nonlinear operator equations
both function and derivative approximations must be taken into account. As
a consequence, inexact Newton methods in Banach spaces are the correct
theoretical frame to study convergence of algorithms—to be treated below in
Sections 7.4, 8.1, and 8.3.
Theorem 2.1 Let F : D → Y be a continuously Fréchet differentiable mapping with D ⊆ X open and convex. For a starting point x0 ∈ D let F (x0 ) be
invertible. Assume that
F (x0 )−1 F (x0 ) ≤ α ,
0 −1 F (x ) (F (y) − F (x)) ≤ ω 0 y − x
S(x0 , ρ− ) ⊂ D,
h0 := αω 0 ≤ 12 ,
ρ− := 1 −
x, y ∈ D ,
(2.2)
(2.3)
1 − 2h0
ω0 .
Then the sequence {xk } obtained from the ordinary Newton iteration is welldefined, remains in S(x0 , ρ− ), and converges to some x∗ with F (x∗ ) = 0.
For h0 < 12 , the convergence is quadratic.
Proof. Rather than giving the classical 1948 proof [126] of L.V. Kantorovich,
we here sketch an alternative affine covariant proof, which dates back to
T. Yamamoto [202] in 1985.
The proof is by induction starting with k = 0. At iterate xk , let the Fréchet
derivative F (xk ) be invertible. Hence we may require the affine covariant
Lipschitz condition
k −1 F (x )
F (y) − F (x) ≤ ω k y − x
and define an associated first majorant
ω k Δxk ≤ hk .
As a preparation to show that with F (xk ) also the Fréchet derivative
F (xk+1 ) is invertible, we define the operators
Bk+1 := F (xk )−1 F (xk+1 )
and the associated second majorant
2.1 Error Oriented Algorithms
47
−1
Bk+1
≤ βk+1 .
Consequently, for k > 0 we have the upper bound
ω k ≤ βk ω k−1 .
By means of the operator perturbation lemma, we easily obtain
βk+1 = 1/(1 − hk ) .
(2.4)
Next, in order to exploit the above Lipschitz condition, we apply standard
analytical techniques to obtain
xk+1 − xk = F (xk )−1
1
F (xk−1 + sΔxk−1 ) − F (xk−1 ) Δxk−1 ds ,
s=0
which implies
ω k xk+1 − xk ≤ 12 βk2 h2k−1 =: hk .
(2.5)
Combination of the two relations (2.5) and (2.4) then yields the single recursive equation
1 2
h
2 k−1
hk =
.
(1 − hk−1 )2
Herein contraction occurs, if
1
2 h0
(1 − h0 )2
< 1,
which directly leads to h0 < 12 . Under this assumption, the convergence is
quadratic.
Things are more complicated for the limiting case h0 = 12 , which requires
extra consideration. In this case, we obtain hk = 12 , k = 1, 2, . . ., which
implies
β k = 2 , ω k ≤ 2k ω 0 .
Insertion into the majorant inequality (2.5) then leads to
lim ω k xk+1 − xk ≤ lim 2k ω 0 xk+1 − xk ≤
k→∞
k→∞
1
2
,
which verifies that
lim xk+1 − xk ≤ lim
k→∞
1
k→∞ 2k−1 ω 0
In the latter case, the convergence is linear.
= 0.
√
Remark 2.1 If we define t∗∗ = 1 + 1 − 2h0 , ρ+ = t∗∗ /ω0 , and assume
that S̄(x0 , ρ+ ) ⊂ D, the solution x∗ can be shown to be unique in S(x0 , ρ+).
The corresponding proof is omitted here.
48
2 Systems of Equations: Local Newton Methods
Bibliographical Note. The name ‘Newton-Kantorovich theorem’ has
been coined, since historically L.V. Kantorovich was probably the first to
prove convergence for Newton’s method in Banach spaces. In 1939, he actually showed linear convergence (see [125]), but not earlier than 1948 he published his famous proof of quadratic convergence (see [126]). Even though this
early theorem has already been phrased in affine covariant terms, nearly all
(with few exceptions) of his later published versions lack this desirable property (see, e.g., the book by L.V. Kantorovich and G. Akhilov [127]). In 1949,
I. Mysovskikh [155] presented an alternative meanwhile classical convergence
theorem, which today is called ‘Newton-Mysovskikh theorem’. That theorem
was not affine invariant in any sense; the following Theorem 2.2 is an affine
covariant version of it. In 1970, an interesting theorem for local convergence
of Newton’s method, already in affine covariant formulation, has been proved
by H.B. Keller in [129], under the relaxed assumption of Hölder continuity
of F (x)—see Exercise 2.4. Since then a huge literature concerning different
aspects of the classical theorems has unfolded, typically in not affine invariant
form—compare, e.g., the monograph of F.A. Potra and V. Pták [171].
Not earlier than 1979, affine invariance as a subject of its own right within
convergence analysis has been emphasized by P. Deuflhard and G. Heindl
in [76]; this paper included an affine covariant (then called affine invariant) rephrasing of the classical Newton-Kantorovich and Newton-Mysovskikh
theorem and permitted a new local convergence theorem for Gauss-Newton
methods for nonlinear least squares problems—see Section 4.3.1. Also around
that time H.G. Bock [29, 31, 32] adopted affine invariance and slightly weakened the Lipschitz condition in the affine covariant Newton-Mysovskikh theorem that had been given in [76]. Following the affine invariance message of
[76], T. Yamamoto has introduced affine covariance into his subtle convergence estimates for Newton’s method—see, e.g., his starting paper [202] and
work thereafter. Later on, the earlier convergence theorem due to L.B. Rall
[174], proved under the assumptions
F (x∗ )−1 ≤ β∗ , F (x) − F (y) ≤ γx − y
has been put into an affine covariant form by G. Bader [15]. For the improved
variant of Rall’s theorem due to W.C. Rheinboldt [176] see Exercise 2.5.
Throughout the subsequent convergence analysis for local Newton-type methods, we will mostly study extensions of the Newton-Mysovskikh theorem
[155], which have turned out to be an extremely useful basis for the construction of algorithms. The subsequent Theorem 2.3 is the ‘refined NewtonMysovskikh theorem’ due to P. Deuflhard and F.A. Potra [82], which has no
classical predecessor, since it relies on affine covariance in its proof.
Next, we present an affine covariant Newton-Mysovskikh theorem. In what
follows, we will return to the case of finite dimensional nonlinear equations,
i.e. to F : D ⊂ Rn → Rn .
2.1 Error Oriented Algorithms
49
Theorem 2.2 Let F : D → Rn be a continuously differentiable mapping with
D ⊂ Rn convex. Suppose that F (x) is invertible for each x ∈ D. Assume that
the following affine covariant Lipschitz condition holds:
−1 F (z)
F (y) − F (x) (y − x) ≤ ωy − x2
for collinear x, y, z ∈ D. For the initial guess x0 assume that
h0 := ωΔx0 < 2
and that S̄(x0 , ρ) ⊂ D for ρ =
(2.6)
Δx0 .
1 − 12 h0
Then the sequence {xk } of ordinary Newton iterates remains in S(x0 , ρ) and
converges to a solution x∗ ∈ S(x0 , ρ). Moreover, the following error estimates
hold
xk+1 − xk ≤ 12 ωxk − xk−1 2 ,
xk − x∗ ≤
xk − xk+1 .
1 − 12 ωxk − xk+1 (2.7)
(2.8)
Proof. First, the ordinary Newton iteration is used for k and k − 1:
Δxk = F (xk )−1 F (xk ) − F (xk−1 ) + F (xk−1 )Δxk−1 .
Application of the above Lipschitz condition yields
Δxk ≤ 12 ωΔxk−1 2 ,
which is (2.7). For the purpose of repeated induction, introduce the following
notation:
hk := ω Δxk .
Multiplication of (2.7) by ω then leads to
hk ≤ 12 h2k−1 .
Contraction of the {hk } is obtained, if h0 < 2 is assumed, which is just (2.6).
From this, as in the proofs of the preceding theorems, we have hk < hk−1 <
h0 < 2 so that there exists
lim hk = 0 .
k→∞
A straightforward induction argument shows that
Δxl ≤
Hence
1
2
hk
l−k
Δxk for l ≥ k .
50
2 Systems of Equations: Local Newton Methods
∞
xl+1 − xk ≤ Δxl + · · · + Δxk ≤ Δxk 1
h
2 k
j=0
j
=
Δxk .
1 − 12 hk
The special case k = 0 implies that all Newton iterates remain in S(x0 , ρ).
Moreover, the results above show that {xk } is a Cauchy sequence, so it converges to some x∗ ∈ S(x0 , ρ). Taking the limit l → ∞ on the previous estimate
yields (2.8). Finally, with ω < ∞ from (2.6) we have that x∗ is a solution
point.
The following theorem has been named ‘refined Newton-Mysovskikh theorem’
in [82].
Theorem 2.3 Let F : D → Rn be a continuously differentiable mapping with
D ⊂ Rn open and convex. Suppose that F (x) is invertible for each x ∈ D.
Assume that the following affine covariant Lipschitz condition holds
−1 F (x)
F (y) − F (x) (y − x) ≤ ωy − x2
for x, y, ∈ D. Let F (x) = 0 have a solution x∗ .
For the initial guess x0 assume that S̄(x∗ , x0 − x∗ ) ⊂ D and that
ωx0 − x∗ < 2 .
(2.9)
Then the ordinary Newton iterates defined by (2.1) remain in the open ball
S(x∗ , x0 − x∗ ) and converge to x∗ at an estimated rate
xk+1 − x∗ ≤ 12 ωxk − x∗ 2 .
(2.10)
Moreover, the solution x∗ is unique in the open ball S(x∗ , 2/ω).
Proof. We define ek := xk − x∗ and proceed for λ ∈ [0, 1] as follows:
xk + λΔxk − x∗ =ek − λF (xk )−1 F (xk ) − F (x∗ ) =F (xk )−1 λ F (x∗ ) − F (xk ) + F (xk )ek 1
=(1 − λ)ek + λF (xk )−1
(F (xk + sek ) − F (xk ))ek ds
s=0
≤(1 − λ)ek + λ2 ωek 2 .
For the purpose of repeated induction assume that ωek ≤ ωe0 < 2 so
that xk ∈ D is guaranteed. Then the above estimate can be continued to
supply
xk + λΔxk − x∗ < (1 − λ)ek + λek = ek ≤ e0 .
From this, any statement xk + λΔxk ∈S(x∗ , x0 − x∗ ) would lead to a contradiction. Hence, xk+1 ∈ D and
2.1 Error Oriented Algorithms
51
ek+1 ≤ 12 ωek 2 ,
which is just (2.10). In order to prove uniqueness in S(x∗ , 2/ω), let x0 := x∗∗
for some x∗∗ = x∗ with F (x∗∗ ) = 0, which implies x1 = x∗∗ as well. Insertion
into (2.10) finally yields the contradiction
x∗∗ − x∗ ≤ ω/2 x∗∗ − x∗ 2 < x∗∗ − x∗ .
This completes the proof.
In view of actual computation, we may combine the results of Theorem 2.2
and 2.3: if we require hk ≤ 1, then contraction towards x∗ shows up, since
1
h
xk+1 − x∗ k
∗
1
2 k
≤
ωx
−
x
≤
≤ 1.
2
k
∗
x − x 1 − 12 hk
Convergence monitor. We are now ready to exploit both convergence
theorems for actual implementation of Newton’s method. First, we define the
contraction factors
Δxk+1 Θk :=
,
Δxk which in terms of the unknown theoretical quantities hk are known to satisfy
Θk =
hk+1
≤ 12 hk < 1 .
hk
(2.11)
Whenever Θk ≥ 1, then the ordinary Newton iteration is classified as ‘not
convergent’.
Computational Kantorovich estimates. Obviously, the assumption h0 ≤
1 implies Θ0 ≤ 1/2. We define the computationally available a-posteriori
estimates
[hk ]1 = 2Θk ≤ hk , k = 0, 1, . . .
and, recalling hk+1 = Θk hk and shifting the index k + 1 → k, also corresponding a-priori estimates
2
[hk ] := Θk−1 [hk−1 ]1 = 2Θk−1
≤ hk ,
k = 1, 2, . . . .
Bit counting lemma. The relative accuracy of these estimates is considered
in the following lemma, the type of which will appear repeatedly in different
context.
Lemma 2.4 Assume that the just introduced Kantorovich estimates [hk ] satisfy the relative accuracy requirement
0≤
hk − [hk ]
≤ σ < 1 , k = 0, 1, . . . .
[hk ]
Then
Θk+1 ≤ (1 + σ)Θk2 , k = 0, 1, . . . .
52
2 Systems of Equations: Local Newton Methods
Proof. We collect the above relations to obtain
Θk+1 ≤ 12 hk+1 ≤ 12 (1 + σ)[hk+1 ] = (1 + σ)Θk2 .
Restricted convergence monitor. With σ → 1 we then end up with
2
Θk ≤ 2Θk−1
,
k = 0, 1, . . . ,
which leads us to the requirement
Θk ≤
1
2
,
k = 0, 1, . . . ,
(2.12)
a convergence criterion more restrictive than (2.11) above. Otherwise we diagnose divergence of the ordinary Newton iteration.
Termination criterion. A desirable criterion to terminate the iteration
would be
xk − x∗ ≤ XTOL ,
(2.13)
with XTOL a user prescribed error tolerance. In view of (2.8) and with
2
hk → [hk ] = 2Θk−1
we will replace this condition by its cheaply computable
substitute
Δxk ≤ XTOL .
(2.14)
2
1 − Θk−1
Note that XTOL can be chosen quite relaxed here, since xk+1 = xk + Δxk is
cheaply available with an accuracy of O(XTOL2 ).
2.1.2 Simplified Newton method
Consider the simplified Newton iteration as introduced above:
k
k
F (x0 )Δx = −F (xk ) , xk+1 = xk + Δx , k = 0, 1, . . . .
(2.15)
Convergence analysis. We study the influence of the fixed initial Jacobian on the convergence behavior. The theorems to be derived are slight improvements of well-known theorems of J.M. Ortega and W.C. Rheinboldt—
see [163].
Theorem 2.5 Let F : D → Rn be a continuously differentiable mapping
with D ⊂ Rn open and convex. Let x0 ∈ D denote a given starting point so
that F (x0 ) is invertible. Assume the affine covariant Lipschitz condition
F (x0 )−1 F (x) − F (x0 ) ≤ ω0 x − x0 (2.16)
2.1 Error Oriented Algorithms
for all x ∈ D. Let
0
h0 := ω0 Δx ≤
1
2
and define
t∗ = 1 −
1 − 2h0 , ρ =
53
(2.17)
t∗
.
ω0
Moreover, assume that S̄(x0 , ρ) ⊂ D. Then the simplified Newton iterates
(2.15) remain in S̄(x0 , ρ) and converge to some x∗ with F (x∗ ) = 0. The
convergence rate can be estimated by
xk+1 − xk ≤ 12 tk + tk−1 , k = 1, 2, . . .
k
k−1
x − x
and
xk − x∗ ≤
with t0 = 0 and
t∗ − tk
, k = 0, 1, . . .
ω0
(2.18)
(2.19)
tk+1 = h0 + 12 t2k , k = 0, 1, . . . .
Proof. We follow the line of the proofs in [163] and use (2.16) to obtain
xk+1 − xk ≤ 12 ω0 xk − xk−1 xk−1 − x0 + xk − x0 .
(2.20)
The result is slightly more complicated than for the ordinary Newton iteration. We therefore turn to a slightly more sophisticated proof technique by
introducing the majorants
ω0 xk+1 − xk ≤ hk , ω0 xk − x0 ≤ tk
with initial values t0 = 0, h0 ≤ 12 . Because of
xk+1 − x0 ≤ xk − x0 + xk+1 − xk and
ω0 xk+1 − xk ≤ 12 hk−1 (tk + tk−1 ) =: hk
we select the two majorant equations
tk+1 = tk + hk , hk = 12 hk−1 tk + tk−1 ,
which can be combined to a single equation of the form
tk+1 − tk = (tk − tk−1 ) tk−1 + 12 (tk − tk−1 ) = 12 (t2k − t2k−1 ) .
Rearrangement of this equation leads to
tk+1 − 12 t2k = tk − 12 t2k−1 .
54
2 Systems of Equations: Local Newton Methods
Since here the right hand side is just an index shift (downward) of the left
hand side, we can apply the so-called Ortega trick to obtain
tk+1 − 12 t2k = t1 − 12 t20 = h0 ,
which may be rewritten as the simplified Newton iteration
tk+1 − tk = −
g(tk )
= g(tk )
g (t0 )
for the scalar equation
g(t) = h0 − t + 12 t2 = 0 .
g
h0
t∗
t∗∗
1
t
Fig. 2.1. Ortega trick: simplified Newton iteration.
As can be seen from Figure 2.1, the iteration starting at t = 0 will converge to
the root t∗ , which exists, if the above quadratic equation has two real roots.
This implies the necessary condition h0 ≤ 1/2, which has been imposed
above. Also from Figure 2.1 we immediately see that g(tk+1 ) < g(tk ), which
is equivalent to hk+1 < hk . Moreover with
tk ≤ t∗ = 1 −
1 − 2h0 ,
we immediately have
xk ∈ S̄(x0 , ρ) ⊂ D .
Hence, for the solution x∗ we also get x∗ ∈ S̄(x0 , ρ). As for the convergence
rates, just observe that
ω0 xk − x∗ ≤
∞
hi = t∗ − tk
i=k
and use (2.20) to verify the remaining statements of the theorem.
2.1 Error Oriented Algorithms
55
Convergence monitor. From Theorem 2.5 we derive that
k+1
Θk =
Δx
k
Δx ≤
hk+1
= 12 (tk+1 + tk ) .
hk
With t0 = 0, t1 = h0 , the condition h0 ≤ 1/2 induces the condition
1
Θ0 =
Δx 0
Δx ≤ 12 h0 ≤
1
4
,
(2.21)
which characterizes the local convergence domain of the simplified Newton
method. In comparison with Θ0 < 1 for the ordinary Newton method, where
a new Jacobian is used at each step, this is a clear reduction. The above
result also shows that the convergence rate may slow down to
Θk < t∗ = 1 −
1 − 2h0 .
We may replace the theoretical quantity t∗ by its computationally available
bounds
[t∗ ] = 1 − 1 − 4Θ0 ≤ 1 − 1 − 2h0 = t∗ ≤ 1 .
Then divergence of the simplified Newton iteration will be defined to occur
when Θk ≥ [t∗ ].
Termination criterion. From (2.19) we may derive the upper bound
xk − x∗ ≤
t∗ − tk
.
ω0
This line is just a different form of the repeated triangle inequality used in
the proof so that
xk − x∗ ≤
∞
j
Δx .
j=k
This gives rise to the upper bound
k
xk − x∗ ≤ Δx (1 + Θk + Θk+1 Θk + . . .) ≤
k
Δx .
1 − t∗
Upon insertion of the estimate [t∗ ] ≤ t∗ from above, we are led to the approximate termination criterion
k
Δx √
≤ XTOL ,
1 − 4Θ0
where XTOL is the user prescribed final error tolerance. Of course, the application of such a criterion will require to start with some Θ0 < 14 .
56
2 Systems of Equations: Local Newton Methods
2.1.3 Newton-like methods
Consider a rather general Newton-like iteration of the form
M (xk )δxk = −F (xk ) , xk+1 = xk + δxk ,
k = 0, 1, . . . .
(2.22)
Convergence analysis. From the basic construction idea, such an iteration
will converge, if M (x) is a ‘sufficiently accurate’ approximation of F (x). The
question will be how to measure the approximation quality and to quantify
the vague term ‘sufficiently accurate’.
Theorem 2.6 Let F : D → Rn be a continuously differentiable mapping with
D ⊂ Rn open and convex. Let M denote an approximation of F . Assume
that one can find a starting point x0 ∈ D with M (x0 ) invertible and constants
α, ω0 , δ0 , δ1 , δ2 ≥ 0 such that for all x, y ∈ D
M (x0 )−1 F (x0 ) ≤ α ,
M (x0 )−1 F (y) − F (x) ≤ ω 0 y − x ,
M (x0 )−1 F (x) − M (x) ≤ δ0 + δ1 x − x0 ,
M (x0 )−1 M (x) − M (x0 ) ≤ δ2 x − x0 ,
2ασ
≤ 1,
(1 − δ0 )2
√
1+ 1−h .
δ0 < 1, σ := max(ω 0 , δ1 + δ2 ), h :=
S(x0 , ρ) ⊂ D with ρ :=
2α
1 − δ0
(2.23)
Then the sequence {xk } generated from the Newton-like iteration (2.22) is
well-defined, remains in S(x0 , ρ) and converges to a solution point x∗ with
F (x∗ ) = 0. With the notation
h :=
ω0
2α
h, ρ± =
σ
1 − δ0
1∓
1−h
the solution x∗ ∈ S(x0 , ρ− ) is unique in
S(x0 , ρ) ∪ D ∩ S(x0 , ρ+ ) .
Proof. For the usual induction proof, the following majorants are convenient
ω 0 δxk ω 0 xk − x0 ≤ hk ,
≤ tk ,
h0
t0
:= αω 0 ,
:= 0 ,
together with
tk+1 = tk + hk .
(2.24)
2.1 Error Oriented Algorithms
57
Proceeding as in the proofs of the preceding theorems, one obtains
xk+1 − xk = M (xk )−1 F (xk )
= M (xk )−1 F (xk ) − F (xk−1 ) + M (xk−1 )(xk − xk−1 ) ≤ M (xk )−1 F (xk ) − F (xk−1 ) − F (xk−1 )(xk − xk−1 ) + M (xk )−1 F (xk−1 ) − M (xk−1 ) (xk − xk−1 ) .
The perturbation lemma yields:
M (xk )−1 M (x0 ) ≤ 1
(1 − δ2 xk − x0 ) .
Combining these intermediate results and using δ i := δi /ω 0 , i = 1, 2, then
supplies
ω 0 xk+1 − xk ≤
1
1 − δ 2 tk
1 2
h
2 k−1
+ δ0 + δ 1 tk−1 hk−1 .
Thus one ends up with the second majorant equation
hk = δ0 + δ 1 tk−1 hk−1 + 12 h2k−1
1 − δ 2 tk .
Reformulation in view of a possible application of the Ortega technique leads
to
1 − δ 2 tk hk − 1 − δ 2 tk−1 hk−1
(2.25)
= 12 t2k − t2k−1 − (1 − δ0 ) (tk − tk−1 )
+ δ 1 + δ 2 − 1 tk tk−1 − t2k−1 .
Obviously, this technique is only applicable, if one requires that
δ 1 + δ 2 = 1 , i.e., δ1 + δ2 = ω0 ,
which will not be the case in general. However, by defining σ as in assumption
(2.23) and redefining
σδxk ≤
hk ,
h0 := ασ ,
0
σx − x ≤
tk ,
t0 := 0 ,
δi
:=
δi /σ ,
i = 1, 2
k
the disturbing term in (2.25) will vanish. Insertion of (2.24) then gives
1 − δ 2 tk (tk+1 − tk ) + (1 − δ0 ) tk − 12 t2k = t1 = ασ ,
which can be rewritten in the form
58
2 Systems of Equations: Local Newton Methods
tk+1 − tk =
h0 − (1 − δ0 )tk + 12 t2k
1 − δ 2 tk
.
This iteration can be interpreted as a Newton-like iteration in R1 for the
solution of
g(t) := h0 − (1 − δ0 )t + 12 t2 = 0 .
The associated two roots
t
∗
t
∗∗
=
(1 − δ0 ) 1 −
are real if
=
(1 − δ0 ) 1 +
2ασ
1−
(1 − δ0 )2
2ασ
1−
(1 − δ0 )2
,
2ασ
≤ 1.
(1 − δ0 )2
This is just assumption (2.23). The remaining part of the proof essentially
follows the lines of the proof of Theorem 2.5 and is therefore omitted here. The above theorem does not supply any direct advice towards algorithmic
realization. In practical applications, however, additional structure on the
approximations M (x) will be given—often as a dependence on an additional
parameter, which can be manipulated in such a way that convergence criteria can be met. A typical version of Newton-like methods is the deliberate
dropping of ‘weak couplings’ in the derivative, which can be neglected on
the basis of insight into the specific underlying problem. In finite dimensions,
deliberate ‘sparsing’ can be used, which means dropping of ‘small’ entries in
a large Jacobian matrix; this technique works efficiently, if the vague term
‘small’ can be made sufficiently precise from the application context. Needless
to say that ‘sparsing’ nicely goes with sparse matrix techniques.
2.1.4 Broyden’s ‘good’ rank-1 updates
In order to derive an error oriented quasi-Newton method, we start by rewriting the secant condition (1.17) strictly in affine covariant terms of quantities
in the domain space of F . This leads to
Ek (J) δxk = δxk+1 = −Jk−1 Fk+1
in terms of the affine covariant update change matrix
Ek (J) := I − Jk−1 J .
Any Jacobian rank-1 update of the kind
2.1 Error Oriented Algorithms
59
δxk+1 v T
J˜k+1 = Jk I − T
, v ∈ Rn , v = 0
v δxk
with v some vector in the domain space of F will both satisfy the secant
condition and exhibit the here desired affine covariance property. The update
with v = δxk is known in the literature as ‘good Broyden update’ [40].
Auxiliary results. The following theorem will collect a bunch of useful
results for a single iterative step of the thus defined quasi-Newton method.
Theorem 2.7 In the notation just introduced, let
δxk+1 δxTk
Jk+1 = Jk I −
δxk 22
(2.26)
denote an affine covariant Jacobian rank-1 update and assume the local contraction condition
δxk+1 2
Θk =
< 12 .
δxk 2
Then:
(I) The update matrix Jk+1 is a least change update in the sense that
Ek (Jk+1 )2
≤ Ek (J)2 , ∀ J ∈ Sk ,
Ek (Jk+1 )2
≤ Θk .
(II) The update matrix Jk+1 is nonsingular whenever Jk is nonsingular, and
its inverse can be represented in the form
δxk+1 δxTk
−1
Jk+1
= I+
Jk−1
(2.27)
(1 − αk+1 )δxk 22
with
αk+1 =
δxTk δxk+1
<
δxk 22
1
2
.
(III) The next quasi-Newton correction is
−1
δxk+1 = −Jk+1
Fk+1 =
δxk+1
.
1 − αk+1
(IV) Iterative contraction in terms of quasi-Newton corrections shows up as
δxk+1 2
Θk
=
< 1.
δxk 2
1 − αk+1
60
2 Systems of Equations: Local Newton Methods
Proof. For the rank-1 update we directly have
Ek (Jk+1 ) =
δxk+1 δxTk
=⇒ Ek (Jk+1 )2 ≤ Θk .
δxk 22
As for the least change update property, we obtain
T δxk+1 δxTk = Ek (J) δxk δxk ≤ Ek (J)2 ,
Ek (Jk+1 )2 = δxk 2 δxk 22 2
2
2
which confirms statement I. By application of the Sherman-Morrison formula
(see, for instance, the book of A.S. Householder [121]), we directly verify the
statements II and III. In order to show IV, we apply the Cauchy-Schwarz
inequality to see that
|αk+1 | ≤ Θk ,
which, for Θk < 1/2, implies
δxk+1 Θk
Θk
=
≤
< 1.
δxk 1 − αk+1
1 − Θk
Algorithmic realization. The result (2.27) may be rewritten as
δxk+1 δxTk
−1
Jk+1 = I +
Jk−1 .
δxk 22
This recursion cannot be used directly for the computation of δxk+1 . However,
the product representation
T
δx
δx
δx1 δxT0
k
k−1
−1
Jk = I +
· ··· · I +
J0−1 .
δxk−1 22
δx0 22
can be applied up to the correction δxk . This consideration leads to a rather
economic recursive ‘good’ Broyden algorithm, which has been used for quite
a while in the public domain code NLEQ1 [161]. It essentially requires the
kmax + 1 quasi-Newton corrections δx0 , . . . , δxk as extra array storage.
Discrete norms for differential equations. Inner products u, v other
than the Euclidean inner product uT v may be used in view of the underlying
problem—such as (discrete) Sobolev inner products for discretized differential
equations. By all means, scaling in the domain space of F should be carefully
considered. This means that any corrections δx arising in the above inner
products should actually be implemented as D−1 δx with appropriate diagonal
scaling matrix D. If D is chosen in agreement with a relative error concept,
then in this way scaling invariance of the algorithm can be assured.
2.1 Error Oriented Algorithms
61
Condition number monitor. Recursive implementations based on the
above rank-1 factorization have often been outruled with the argument that
some hidden ill-conditioning in the arising Jacobian updates might occur. In
order to derive a some monitor, we may use
δxk+1 δxTk
cond2 (Jk+1 ) ≤ cond2 I +
cond2 (Jk ) .
δxk 22
In this context, the following technical lemma may be helpful.
Lemma 2.8 Given a rank-1 matrix
A=I−
uv T
u2
with Θ :=
< 1,
vT v
v2
its condition number can be bounded as
cond2 (A) ≤
1+Θ
.
1−Θ
Proof. We just use the two bounds
T
uv A ≤ 1 + v T v ≤ 1 + Θ,
A
−1
T −1
uv −1
≤ 1−
≤ (1 − Θ)
vT v and insert into the definition cond2 (A) = A A−1 .
With this result and Θk < 1/2 we are certainly able to assure that
cond2 (Jk+1 ) ≤
1 + Θk
cond2 (Jk ) < 3 cond2 (Jk ) .
1 − Θk
Convergence monitor. In accordance with the above theoretical results,
we impose the condition Θk < 1/2 throughout the whole iteration. Note
that this is an extension of the local convergence domain compared with the
simplified Newton method where Θ0 ≤ 1/4 has to be required. With these
preparations we are now ready to state the ‘good Broyden algorithm’ QNERR
(for ERRor oriented Quasi-Newton method) in the usual informal manner.
Algorithm QNERR.
For given
x0 :
F0 = F (x0 )
evaluation and store
J0 δx0 = −F0
linear system solve
σ0 =
δx0 22
store δx0 , σ0
62
2 Systems of Equations: Local Newton Methods
For k = 0, . . . , kmax :
I.
xk+1 = xk + δxk
Fk+1 = F (x
k+1
new iterate
)
evaluation
J0 v = −Fk+1
II.
linear system solve
If k > 0: for i = 1, . . . , k
α :=
vT δxi−1
,
σi−1
v := v + αδxi
III.
Compute
αk+1
v T δxk
:=
, Θk =
σk
vT v
σk
1/2
If Θk > 12 : stop, no convergence
v
IV. δxk+1 =
,
1 − αk+1
σk+1 = δxk+1 22
If
√
store
store
store
σk+1 ≤ XTOL:
solution x∗ = xk+1 + δxk+1
Else:
no convergence within kmax iterations.
Convergence analysis. The above Theorem 2.7 does not give conditions,
under which the contraction condition Θk < 1/2 is assured throughout the
whole iteration. This will be the topic of the next theorem.
Theorem 2.9 For F : D −→ Rn be a continuously differentiable mapping
with D open and convex. Let x∗ ∈ D denote a unique solution point of F
with F (x∗ ) nonsingular. Assume that the following affine covariant Lipschitz
condition holds:
F (x∗ )−1 F (x) − F (x∗ ) v ≤ ωx − x∗ · v
for x, x + v ∈ D and 0 ≤ ω < ∞. Consider the quasi-Newton iteration as
defined in Theorem 2.7. For some Θ in the range 0 < Θ < 1 assume that:
(I) the initial approximate Jacobian J0 satisfies
δ0 := F (x∗ )−1 J0 − F (x0 ) < Θ/(1 + Θ) ,
(2.28)
2.1 Error Oriented Algorithms
63
(II) the initial guess x0 satisfies
1−Θ
t0 := ωx − x ≤
2−Θ
∗
0
Θ
− δ0 .
1+Θ
(2.29)
Then the quasi-Newton iterates {xk } converge to x∗ in terms of errors as
xk+1 − x∗ < Θxk − x∗ ,
(2.30)
or, in terms of corrections as
δxk+1 ≤ Θδxk .
(2.31)
The convergence is superlinear with
δxk+1 = 0.
k→∞ δxk lim
As for the Jacobian rank-1 updates, the ‘bounded deterioration property’ holds
in the form
Θ
Ek := F (x∗ )−1 Jk − I ≤
< 12
(2.32)
1+Θ
together with the asymptotic property
lim
k→∞
Ek δxk = 0.
δxk (2.33)
Proof. Let · be · 2 throughout. For ease of writing we characterize the
Jacobian update approximation by
(T )
ηk =
Ek v
Ek δxk , η k = Ek = EkT = max
.
v
=
0
δxk v
By definition, ηk ≤ η k . For the convergence analysis we introduce
tk = ωek , ek := xk − x∗ .
As usual [57], the proof is performed in two basic steps: first linear convergence, then superlinear convergence.
I. To begin with, exploit the Lipschitz condition in the form
F (x∗ )−1 F (xk+1 )
≤
1 ∗ −1 k
F (x )
F (x + sδxk ) − F (x∗ ) δxk ds
s=0
+Ek δxk ≤
1
2 (tk+1
+ tk ) + ηk δxk .
64
2 Systems of Equations: Local Newton Methods
Under the assumption ηk+1 < 1 we may estimate
F (x∗ )−1 F (xk+1 ) = (I + Ek+1 )δxk+1 ≥ (1 − ηk+1 )δxk+1 so that
δxk+1 η k + tk
≤
, where tk := 12 (tk + tk+1 ) .
δxk 1 − ηk+1
(2.34)
As for the iterative errors ek , we may derive the relation
ek+1
=
=
ek − Jk−1 F (xk )
−1
(I + Ek )
1 ∗
∗ −1
∗
Ek ek − F (x )
F (x + sek ) − F (x ) ek ds ,
s=0
from which we obtain the estimate ( let η k < 1)
tk+1 ≤
η k + 12 tk
tk .
1 − ηk
(2.35)
Upon comparing the right hand upper bounds in (2.35) and (2.34) we are led
to define the majorant
η + tk
Θ := k
,
(2.36)
1 − ηk
which implies that
tk+1 < Θtk .
(2.37)
II. Next, we study the approximation properties of the Jacobian updates.
With Ek as defined, the above rank-1 update may be rewritten in the form
Ek+1 = Ek + F (x∗ )−1
If we insert
Fk+1 δxTk
.
δxk 22
F (x∗ )−1 Fk+1 = (Dk+1 − Ek )δxk ,
wherein
1
∗ −1
Dk+1 := F (x )
k
F (x + sδxk ) − F (x∗ ) ds
s=0
and introduce the orthogonal projections
Q⊥
k = I − Qk =
δxk δxTk
,
δxk 2
then we arrive at the decomposition
Ek+1 = Ek Qk + Dk+1 Q⊥
k
2.1 Error Oriented Algorithms
65
and its transpose (v = 0 arbitrary)
Note that
T
T
Ek+1
v = Qk EkT v + Q⊥
k Dk+1 v .
(2.38)
Ek+1 δxk Dk+1 δxk =
≤ tk .
δxk δxk (2.39)
III. In order to prove linear convergence, equation (2.38) is used for the quite
rough estimate
η k+1 = max
v=0
T
Ek+1
v
EkT v
| Dk+1 δxk , v|
≤ max
+max
≤ η k +tk . (2.40)
v=0
v=0
v
v
δxk v
Assume now that we have uniform upper bounds
Θ ≤ Θ < 1 , ηk ≤ η < 1 .
Then (2.37) can be replaced by
tk+1 < Θtk < tk
and (2.36) leads to the natural definition
Θ≤
η + t0
=: Θ .
1−η
(2.41)
As for the definition of η, we apply (2.40) to obtain
k
η k+1 < η 0 +
tl < η 0 +
l=0
t0
=: η .
1−Θ
(2.42)
Insertion of η into (2.41) then eventually yields after some calculation:
Θ
t0 ≤ (1 − Θ)
− η0 ,
(2.43)
1+Θ
which obviously requires
η0 <
Θ
<
1+Θ
1
2
for Θ < 1 .
Observe now that by mere triangle inequality, with δ0 as defined in (2.28), we
have η 0 ≤ t0 + δ0 . Therefore, the assumption (2.43) can finally be replaced
by the above two assumptions (2.28) and (2.29). Once such a Θ < 1 exists,
we have (2.30) directly from tk+1 < Θtk and (2.31) from inserting η into
(2.34). The bounded deterioration property (2.32) follows by construction
and insertion of (2.29) into (2.42).
66
2 Systems of Equations: Local Newton Methods
IV. In order to show superlinear convergence, we use (2.38) in a more subtle
manner. In terms of the Euclidean inner product ·, ·, some short calculation
supplies the equation
T
Ek+1
v2 = EkT v2 −
Ek δxk , v2
Dk+1 δxk , v2
+
.
δxk 2
δxk 2
Summing over the indices k, we arrive at
l
k=0
T
El+1
v2
Ek δxk , v2
E0T v2
=
−
+
2
2
2
v δxk v
v2
l
k=0
Dk+1 δxk , v2
.
δxk 2 v2
Upon dropping the negative right hand term, letting l → ∞, and using (2.39)
with tk+1 < Θ · tk , we end up with the estimate
∞
k=0
Ek δxk , v2
1+Θ 2
≤ η 20 + 12
t0 .
2
2
v δxk 1−Θ
Since the right hand side is bounded, we immediately conclude that
lim
k→∞
Ek δxk , v2
= 0 ∀ v ∈ Rn .
δxk 2 v2
As a consequence, with
ξk :=
δxk
,
δxk we must have
lim Ek ξk = 0
k→∞
from which statement (2.33) follows. Finally, with (2.34), we have proved
superlinear convergence.
Bibliographical Note. Quasi-Newton methods are described, e.g., in the
classical optimization book [57] by J.E. Dennis and R.B. Schnabel or, more
recently, in the textbook [132] by C.T. Kelley. These methods essentially
started with the pioneering paper [40] by C.G. Broyden. For quite a time,
the convergence of the ‘good’ Broyden method was not at all clear. A breakthrough in its convergence analysis came by the paper [41] of C.G. Broyden,
J.E. Dennis, and J.J. Moré, where local and superlinear convergence has been
shown on the basis of condition (2.33), the meanwhile so-called Dennis-Moré
condition (see [55]). To the most part, the present section is an affine covariant reformulation of well-known material spread over a huge literature—see,
e.g., the original papers [56] by J.E. Dennis and R.B. Schnabel or [58] by
J.E. Dennis and H.F. Walker.
The above quasi–Newton algorithm is realized within the earlier code NLEQ1
and its update NLEQ-ERR.
2.1 Error Oriented Algorithms
67
2.1.5 Inexact Newton-ERR methods
Inexact Newton methods consist of a combination of an outer iteration, the
Newton iteration, and an inner iteration such that (dropping the inner iteration index i)
F (xk )(δxk − Δxk ) = rk ,
xk+1 = xk + δxk ,
k = 0, 1, . . . .
Here the inner residual rk gives rise to the difference between the exact Newton correction Δxk and the inexact Newton correction δxk . Among the possible inner iterative solvers we will concentrate on those that reduce the Euclidean error norms δxk − Δxk , which leads us to CGNE (compare Section
1.4.3) and to GBIT (compare Section 1.4.4). In both cases, the perturbation
will be measured by the relative difference between the exact Newton correction Δxk and the inexact Newton correction δxk via
δk =
δxk − Δxk , k = 0, 1, . . . .
δxk (2.44)
As a guiding principle for convergence, we will focus on contraction in terms
of the (not actually computed) exact Newton corrections
Θk =
Δxk+1 ,
Δxk subject to the perturbation coming from the truncation of the inner iteration.
Convergence analysis—CGNE. First we work out details for the error minimizing case, exemplified by CGNE specifying the norm · to be the Euclidean
norm · 2 . Upon recalling (1.28), the starting value δxk0 = 0 for the CGNE
iteration implies that
Δxk = δxk 1 + δk2 ≥ δxk .
Moreover, from (1.29) and (1.30) we conclude that δk is monotonically decreasing in the course of the inner iteration so that eventually any threshold
condition of the type δk ≤ δ̄ can be met. With this preparation, we are now
ready to state our convergence result.
Theorem 2.10 Let F : D −→ Rn be a continuously differentiable mapping
with D ⊂ Rn open, convex, and sufficiently large. Suppose that F (x) is
invertible for each x ∈ D. Assume that the following affine covariant Lipschitz
condition holds:
F (z)−1 F (y) − F (x) v ≤ ωy − x · v
for collinear x, y, z ∈ D .
68
2 Systems of Equations: Local Newton Methods
Let x0 ∈ D denote a given starting point for a Newton-CGNE iteration. At an
iterate xk , let δk as defined in (2.44) denote the relative error of the inexact
Newton correction δxk . Let the inner CGNE iteration be started with δxk0 = 0,
which gives rise to the following relations between the Kantorovich quantities
hk := ωΔxk hδk := ωδxk =
and
hk
1 + δk2
.
Let x∗ ∈ D be the unique solution point.
I. Linear convergence mode. Assume that an initial guess x0 has been
chosen such that
h0 < 2Θ < 2
for some Θ < 1. Let δk+1 ≥ δk be realized throughout the inexact Newton
iteration and control the inner iteration such that
ϑ(hk , δk ) =
1 δ
2 hk
+ δk (1 + hδk )
1 + δk2
≤ Θ,
which assures that
δk ≤ Θ
1−Θ
.
(2.45)
2
Then this implies the exact monotonicity
Δxk+1 ≤Θ
Δxk and the inexact monotonicity
δxk+1 ≤
δxk 1 + δk2
Θ ≤ Θ.
2
1 + δk+1
The iterates {xk } remain in S(x0 , ρ) with ρ = δx0 /(1 − Θ) and converge
at least linearly to x∗ .
II. Quadratic convergence mode. For some ρ > 0, let the initial guess
x0 satisfy
2
h0 <
(2.46)
1+ρ
and control the inner iteration such that
δk ≤
which requires that
ρ hδk
,
2 1 + hδk
(2.47)
2.1 Error Oriented Algorithms
ρ>
3δ0
1 − δ0
69
(2.48)
be chosen. Then the inexact Newton iterates remain in S(x0 , ρ) with
1+ρ
ρ = δx0 / 1 −
h0
2
and converge quadratically to x∗ with
Δxk+1 ≤
1+ρ
ωΔxk 2
2
δxk+1 ≤
1+ρ
ωδxk 2 .
2
and
Proof. First we show that
1
Δx
k+1
≤
k+1 −1 k
F (x )
F (x +tδxk )−F (xk ) δxk dt+F (xk+1 )−1 rk .
t=0
For the first term we just apply the Lipschitz condition in standard form. For
the second term we may use the same condition plus the triangle inequality
to obtain
F (xk+1 )−1 rk = F (xk+1 )−1 F (xk )(δxk − Δxk ) ≤ (1 + hδk )δxk − Δxk .
With definition (2.44), this gives
Δxk+1 ≤ 12 hδk + δk (1 + hδk ) .
δxk (2.49)
With hδk = hk / 1 + δk2 we then arrive at
Δxk+1 ≤ ϑ(hk , δk )) =
Δxk 1 δ
2 hk
+ δk (1 + hδk )
1 + δk2
.
In order to prove linear convergence, we might require ϑ(hk , δk ) = Θ < 1,
which implies that δk monotonically increases as hk monotonically decreases—
which would automatically lead to δk+1 ≥ δk when hk+1 ≤ hk . However, since
strict equality cannot be realized within CGNE, we have to assume the two
separate inequalities ϑ ≤ Θ and δk+1 ≥ δk , as done in the theorem. Note that
a necessary condition for ϑ(hk , δk ) ≤ Θ with some δk > 0 is that it holds at
least for δk = 0, which yields h0 < 2Θ, the assumption made in the theorem.
As for the contraction in terms of the inexact Newton corrections, we then
obtain
70
2 Systems of Equations: Local Newton Methods
δxk+1 =
δxk 1 + δk2 Δxk+1 ≤
2
1 + δk+1
Δxk 1 + δk2
Θ ≤ Θ.
2
1 + δk+1
Usual linear convergence results then imply that {xk } remains in S(x0 , ρ)
with ρ = δx0 /(1 − Θ), if only S(x0 , ρ) ⊂ D, which we assumed by D to be
‘sufficiently large’. Asymptotically we thus assure that ϑ(0, δk ) ≤ Θ, which is
equivalent to (2.45).
For the quadratic convergence case we require that the first term in ϑ(hk , δk )
originating from the outer iteration exceeds the second term, which brings
us to (2.47). Note that now hk+1 ≤ hk implies δk+1 ≤ δk and hk → 0 also
δk → 0—a behavior that differs from the linear convergence case. Insertion
of (2.47) into ϑ(hk , δk ) then directly leads to
Δxk+1 1 + ρ hk
1+ρ
≤
≤
hk
Δxk 2 1 + δk2
2
and to
δxk+1 1+ρ
hδ
k
≤
k
δx 2
1 + δ2
≤
1+ρ δ
hk .
2
k+1
Upon applying the usual quadratic convergence results, we have to require
the sufficient condition
1+ρ δ
1+ρ
h0 ≤
h0 < 1
2
2
and then, assuming that D is ‘sufficiently large’, obtain convergence within
the ball
δx0 S(x0 , ρ) , ρ = 1+ρ
1−
h0
2
as stated above. Finally, upon inserting (2.46) into (2.47) and using hδ0 ≤ h0 ,
the result (2.48) is readily confirmed.
Convergence analysis—GBIT. By a slight modification of Theorem 2.10,
the Newton-GBIT iteration can also be shown to converge.
Theorem 2.11 Let δk < 12 in (2.44) and replace the Kantorovich quantities
hδk in Theorem 2.10 by their upper bounds such that
hδk =
Then we obtain the results:
hk
.
1 − δk
2.1 Error Oriented Algorithms
71
I. Linear convergence mode. Let δk in each inner iteration be controlled
such that
1 δ
h + δk (1 + hδk )
ϑ(hk , δk ) = 2 k
≤ Θ,
1 − δk
which assures that
Θ
δk ≤
.
(2.50)
1+Θ
Then this implies the inexact monotonicity test
δxk+1 1 − δk
≤
Θ
k
δx 1 − δk+1
(2.51)
and the exact monotonicity test
Δxk+1 ≤ Θ.
Δxk II. Quadratic convergence mode. Let the inner iteration be controlled
according to (2.47) and
2(1 − δ0 )2
h0 <
.
(2.52)
1+ρ
Then (2.48) needs to be replaced by
ρ>
δ0 (3 − 2δ0 )
.
1 − 2δ0
(2.53)
The exact Newton corrections behave like
Δxk+1 ≤
1
2
1+ρ
ωΔxk 2
(1 − δk )2
and the inexact Newton corrections like
δxk+1 ≤
1
2
1+ρ
ωδxk 2 .
1 − δk+1
Proof. The main difference to the previous theorem is that now we can only
apply the triangle inequality
| Δxk − δxk − Δxk | ≤ δxk ≤ δxk − Δxk + Δxk .
Assuming δk < 1 in definition (2.44), we obtain
Δxk Δxk ≤ δxk ≤
,
1 + δk
1 − δk
which motivates the majorant ωδxk ≤ hδk as stated in the theorem. Upon
revisiting the proof of Theorem 2.10, the result (2.49) is seen to still hold,
which is
72
2 Systems of Equations: Local Newton Methods
Δxk+1 ≤ 12 hδk + δk (1 + hδk ) .
δxk (2.54)
From this, we obtain the modified estimate for the exact Newton corrections
Δxk+1 ≤
Δxk 1 δ
h
2 k
+ δk (1 + hδk )
=
1 − δk
1
h
2 k
+ δk (1 − δk + hk )
= ϑ(hk , δk ) .
(1 − δk )2
In a similar way, we obtain for the inexact Newton corrections
δxk+1 ≤
δxk 1 δ
h
2 k
+ δk (1 + hδk )
1 − δk
=
ϑ(hk , δk ) .
1 − δk+1
1 − δk+1
For the linear convergence mode, we adapt δk such that
ϑ(hk , δk ) ≤ Θ .
Asymptotically we thus assure that ϑ(0, δk ) ≤ Θ, equivalent to (2.50).
For the quadratic convergence mode, we again require (2.47) (with hδk in the
present meaning, of course), i.e.
δk ≤ 12 ρ
hδk
.
1 + hδk
With this choice we arrive at
Δxk+1 ≤
Δxk 1
2
1+ρ δ
h =
1 − δk k
1
2
1+ρ
ωΔxk (1 − δk )2
for the exact Newton contraction, which requires (2.52) as a necessary condition. Upon combining (2.47) and (2.52), we obtain
δ0 ≤ 12 ρ
hδ0
ρ(1 − δ0 )
<
.
δ
1 + ρ + 2(1 − δ0 )
1 + h0
Given ρ, this condition would lead to some uneasy quadratic root. Given δ0 ,
we merely have the linear inequality
ρ>
δ0 (3 − 2δ0 )
,
1 − 2δ0
which is (2.53); it necessarily requires δ0 < 1/2 in agreement with the assumption δk < 1/2 of the theorem.
The corresponding bound for the inexact Newton corrections is
δxk+1 ≤
δxk which completes the proof.
1
2
1+ρ
ωδxk ,
1 − δk+1
2.1 Error Oriented Algorithms
73
Convergence monitor. Assume that the quantity Θ < 1 in the linear
convergence mode or the quadratic convergence mode have been specified;
in view of (2.12), we may require that Θ ≤ 1/2. The desirable convergence
criterion would be
Δxk+1 2
Θk :=
≤ Θ.
Δxk 2
Since this criterion cannot be directly implemented, Θk needs to be substik ≈ Θk .
tuted by a computationally available Θ
For CGNE with δxk0 = 0, this leads to the inexact monotonicity test
2
1 + δ̄k+1
δxk+1 2
k =
Θ
·
≤ Θ,
(2.55)
2
δxk 2
1 + δ̄k
where the quantities δ̄k , δ̄k+1 are the computationally available estimates for
the otherwise unavailable quantities δk , δk+1 as given in (1.32).
For GBIT, the result (2.51) suggests the following inexact monotonicity test
k+1
k = 1 − δ̄k+1 · δx ≤ Θ .
Θ
δxk 1 − δ̄k
(2.56)
As an alternative, we may also consider the weaker necessary condition
k+1
k = 1 − δ̄k+1 · δx ≤ Θk ≤ Θ
Θ
δxk 1 + δ̄k
(2.57)
or the stronger sufficient condition
k+1
k = 1 + δ̄k+1 · δx ≤ Θ
Θk ≤ Θ
δxk 1 − δ̄k
(2.58)
for use within the convergence monitor.
Preconditioning. In order to speed up the inner iteration, preconditioning
from the left or/and from the right may be used. This means solving
−1 k
CL F (xk )CR CR
δx − Δxk = CL rk .
In such a case, we will define
δk =
−1
CR
(Δxk − δxk )
.
−1
CR
δxk Of course, in this case the preconditioned error norm is reduced by the inner
iteration, whereas CL only affects its rate of convergence. Consequently, any
adaptive strategy should then, in principle, be based upon the contraction
factors
−1
CR
Δxk+1 Θk =
−1
CR
Δxk k ≈ Θk as in (2.55) for CGNE or any
and its corresponding scaled estimate Θ
choice between (2.56), (2.57), and (2.58) for GBIT.
74
2 Systems of Equations: Local Newton Methods
Termination criterion. In the same spirit as above, we mimic the termination criterion (2.14) for the exact Newton iteration by requiring for CGNE
the substitute condition
1 + δ̄k2
δxk 2 ≤ XTOL
2
1−Θ
k−1
and for GBIT the sufficient condition
1 + δ̄k
δxk ≤ XTOL ,
2
1−Θ
k−1
each for the finally accepted iterate xk+1 , where XTOL is a user prescribed
absolute error tolerance (to be replaced by some relative or some scaled error
criterion).
Estimation of Kantorovich quantities. In order to deal successfully with
the question of how to match inner and outer iterations, the above theory
obviously requires the theoretical quantities hδk = ωδxk —which, however,
are not directly available. In the spirit of the whole book we aim at replacing
these quantities by computational estimates [hδk ]. Recalling Section 2.1.1, we
2
aim at estimating the a-priori estimates [hk ] = 2Θk−1
≤ hk for k ≥ 1.
For CGNE with initial correction δxk0 = 0, we replace the relative errors δk by
their estimates δ̃k from Section 1.4.3 and thus arrive at the a-priori estimates
2 ≤ hk , k = 1, 2, . . . ,
[hδk ] = [hk ]/ 1 + δ̄k2 , [hk ] = 2Θ
(2.59)
k−1
k−1 from (2.55) is inserted.
where Θ
For GBIT, we get the a-priori estimates
[hδk ] =
[hk ]
,
(1 − δ̄k )
2 ≤ hk ,
[hk ] = 2Θ
k−1
k = 1, 2, . . . ,
(2.60)
k−1 from (2.57) is inserted.
where Θ
In both CGNE and GBIT, we may alternatively use the a-posteriori estimates
k−1
[hk−1 ]1 = 2Θ
and insert them either into (2.59) or into (2.60), respectively, to obtain
[hδk−1 ]1 . From this, we may construct the a-priori estimates (for k ≥ 1)
[hδk ] = [hδk−1 ]1
δxk .
δxk−1 Note that in CGNE this formula inherits the saturation property.
For k = 0, we cannot but choose any ‘sufficiently small’ δ0 —as stated in the
quadratic convergence mode to follow next.
2.1 Error Oriented Algorithms
75
Standard convergence mode. In this mode the inner iteration is terminated whenever
δk ≤ δ̄
(2.61)
for some default value δ̄ < 1 to be chosen. In this case, asymptotic linear
convergence is obtained.
For CGNE, Theorem 2.10 requires
δ̄/ 1 + δ̄ 2 < Θ ,
which for Θ = 12 leads to the restriction δ̄ <
Theorem 2.11 requires
δ̄/(1 − δ̄) < Θ ,
√
3/3 ≈ 0.577. For GBIT,
which leads to δ̄ < 1/3. In any case, we recommend to choose δ̄ ≤ 1/4 to
assure at least two binary digits.
Quadratic convergence mode. In CGNE, we set δ0 = 14 in (2.48) and
obtain ρ > 1—thus assuring at least the first binary digit. In GBIT, we also
set δ0 = 14 and apply the inequality (2.53) thus arriving at ρ > 54 .
As for the adaptive termination of the inner iteration, we want to satisfy
condition (2.47) for k ≥ 1. Following our paradigm, we will replace the computationally unavailable quantity hδk therein by its computational estimate
[hδk ], which yields, for both CGNE and GBIT, the substitute condition
δ̄k ≤ 12 ρ ·
[hδk ]
.
1 + [hδk ]
(2.62)
Whenever δk ≤ δ̄k , the above monotone increasing right side as a function of
[hδk ] and the relation [hδk ] ≤ hδk imply that the theoretical condition (2.47) is
actually assured with (2.62). Based on the a-priori estimates (2.59) or (2.60),
respectively, we obtain a simple nonlinear scalar equation for an upper bound
of δk .
Note that δk → 0 is enforced when k → ∞, which means: the closer the
iterates come to the solution point, the more work needs to be done in the
inner iteration to assure quadratic convergence of the outer iteration.
k
Linear convergence mode. Once the approximated contraction factor Θ
is sufficiently below some prescribed threshold value Θ ≤ 1/2, we may switch
to the linear convergence mode described in either of the above two convergence theorems. As for the termination of the inner iteration, we recall the
theoretical condition
ϑ(hk , δk ) ≤ Θ .
Since the quantity ϑ is unavailable, we will replace it by the computationally
available estimate
76
2 Systems of Equations: Local Newton Methods
[ϑ(hk , δk )] = ϑ([hk ], δk ) ≤ ϑ(hk , δk ) .
As this mode occurs only for k > 0, we can just insert the a-priori estimates
(2.59) or (2.60), respectively. Since the above right hand side is a monotone
increasing function of hk and [hk ] ≤ hk , this estimate may be ‘too small’
and therefore lead to some δk , which is ‘too large’. Fortunately, the difference between computational estimate and theoretical quantity can be ignored
asymptotically. In any case, we require the monotonicity (2.55) for CGNE or
(2.56), (2.57), or (2.58) for GBIT and run the inner iteration at each step k
until either the actual value of δk obtained in the course of the inner iteration
k > 2Θ.
satisfies the condition above or divergence occurs with Θ
In CGNE, we observe that in this mode the closer the iterates come to the
solution point, the less work is necessary within the inner iteration to assure
linear convergence of the outer iteration. In GBIT, this process continues only
until the upper bound (2.50) for δk has been reached.
The here described error oriented local inexact Newton algorithms are self–
contained and similar in spirit, but not identical with the local parts of the
global inexact Newton codes GIANT-CGNE and GIANT-GBIT, which are worked
out in detail in Section 3.3.4 below.
Bibliographical Note. A first affine covariant convergence analysis of
a local inexact Newton method has been given by T.J. Ypma [203]. The
first affine covariant inexact Newton code has been GIANT, developed by
P. Deuflhard and U. Nowak [67, 160] in 1990. That code had also used a
former version of GBIT for the inner iteration.
2.2 Residual Based Algorithms
In most algorithmic realizations of Newton’s method iterative values of the
residual norms are used for a check of convergence. An associated convergence
analysis will start from affine contravariant Lipschitz conditions of the type
(1.8) and lead to results in terms of residual norms only, which are tacitly
assumed to be scaled. As explained in Section 1.2.2 above, such an analysis
will not touch upon the question of local uniqueness of the solution.
2.2.1 Ordinary Newton method
Recall the notation of the ordinary Newton method
F (xk )Δxk = −F (xk ) , xk+1 = xk + Δxk ,
k = 0, 1, . . . .
(2.63)
2.2 Residual Based Algorithms
77
Convergence analysis. Analyzing the iterative residuals leads to an affine
contravariant version of the well-known Newton-Mysovskikh theorem.
Theorem 2.12 Let F : D → Rn be a differentiable mapping with D ⊂ Rn
open and convex. Let F (x) be invertible for all x ∈ D. Assume that the
following affine contravariant Lipschitz condition holds:
F (y) − F (x) (y − x) ≤ ωF (x)(y − x)2 for x, y ∈ D .
Define the open level set Lω = x ∈ D| F (x) < ω2 and let Lω ⊂ D be
bounded. For a given initial guess x0 of an unknown solution x∗ let
h0 := ωF (x0 ) < 2 , i.e. x0 ∈ Lω .
(2.64)
Then the ordinary Newton iterates {xk } defined by (2.63) remain in Lω and
converge to some solution point x∗ ∈ Lω with F (x∗ ) = 0. The iterative
residuals {F (xk )} converge to zero at an estimated rate
F (xk+1 ) ≤ 12 ωF (xk )2 .
(2.65)
Proof. To show that xk+1 ∈ D we apply the integral form of the mean value
theorem and the above Lipschitz condition and obtain
F (xk + λΔxk ) =
=
F (xk ) +
λ λ
F (xk + tΔxk )Δxk dt
t=0
F (xk + tΔxk ) − F (xk ) Δxk
t=0
+(1 − λ)F (xk ) dt
≤
λ
(F (xk + tΔxk ) − F (xk ) Δxk dt
t=0
+(1 − λ)F (xk )
≤
=
λ
ω
F (xk )Δxk 2 t dt + (1 − λ)F (xk )
t=0
1 − λ + 12 ωλ2 F (xk ) F (xk )
for each λ ∈ [0, 1] such that xk + tΔxk ∈ Lω for t ∈ [0, λ]. Now assume that
xk+1 ∈
/ Lω . Then there exists a minimal λ̄ ∈ ]0, 1] with xk + λ̄Δxk ∈ ∂Lω
and F (xk + λ̄Δxk ) < (1 − λ̄ + λ̄2 )F (xk ) < 2/ω, which is a contradiction.
For λ = 1 we get relation (2.65). In terms of the residual oriented so-called
Kantorovich quantities
hk := ωF (xk )
(2.66)
we may obtain the quadratic recursion
hk+1 ≤ 12 h2k = ( 12 hk )hk .
(2.67)
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2 Systems of Equations: Local Newton Methods
With assumption (2.64), h0 < 2, we obtain h1 < h0 < 2 for k = 0 and, by
repeated induction over k, then
hk+1 < hk < 2 , k = 0, 1, . . .
⇒
lim hk = 0 .
k→∞
This can be also written in terms of the residuals as
F (xk+1 ) < F (xk ) <
2
ω
⇒
lim F (xk ) = 0 .
k→∞
In terms of the iterates we have
{xk } ⊂ Lω ⊂ D .
Since Lw is bounded, there exists an accumulation point x∗ of {xk } with
F (x∗ ) = 0, i.e. x∗ is a solution point, but not necessarily unique in Lω . This theorem also holds for underdetermined nonlinear systems—compare
Exercise 4.10.
Convergence monitor. We now want to exploit Theorem 2.12 for actual
computation. For this purpose, we introduce the contraction factors
Θk :=
F (xk+1 )
F (xk )
and write (2.67) in the equivalent form
Θk =
hk+1
≤ 12 hk .
hk
(2.68)
For k = 0, assumption (2.64) assures residual monotonicity
Θ0 < 1 .
(2.69)
Whenever Θ0 ≥ 1, the assumption (2.64) is certainly violated, which means
that the initial guess x0 is not ‘sufficiently close’ to the solution point x∗
in the sense of the above theorem. Suppose now that the test Θ0 < 1 has
been passed. For the construction of a quadratic convergence monitor we introduce computationally available estimates [hk ] for the unknown theoretical
quantities hk from (2.66). In view of (2.68) we may define the computational
a-posteriori estimate
[hk ]1 = 2Θk ≤ hk
and, since hk+1 = Θk hk , also the a-priori estimate
[hk+1 ] = Θk [hk ]1 = 2Θk2 ≤ hk+1 .
2.2 Residual Based Algorithms
79
Upon roughly identifying [hk+1 ]1 ≈ [hk+1 ], we arrive at the approximate
recursion (k = 0, 1, . . .):
Θk+1 ≈ Θk2 ≤ Θ0 < 1 .
Violation of this recursion at least in the mild sense
Θk+1 > Θ0
or the stricter sense
Θk+1 ≥ 2Θk2
may be used to terminate the ordinary Newton iteration as ‘not convergent’.
Termination criterion. This affine contravariant theory agrees with a termination criterion of the form
F (x̂) ≤ FTOL ,
(2.70)
where FTOL is a user prescribed residual error tolerance.
Computational complexity. A short calculation shows that, for a given
starting point x0 , the number q of iterations such that x̂ = xq+1 meets the
above termination requirement satisfies roughly
q ≈ ld
log(FTOL /F (x0 ))
.
log Θ0
(2.71)
The proof is left as Exercise 2.1. In other words, with ‘sufficiently good’ initial
guesses x0 of the solution x∗ at hand, the computational complexity of the
nonlinear problem is comparable to the one of the linearized problem. Such
problems are sometimes called mildly nonlinear .
2.2.2 Simplified Newton method
Recall the notation of the simplified Newton iteration
k
k
F (x0 )Δx = −F (xk ) , xk+1 = xk + Δx , k = 0, 1, . . . .
(2.72)
Convergence analysis. Here we study convergence in terms of iterative
residuals obtaining an affine contravariant variant of the Newton-Kantorovich
theorem—without any uniqueness results, of course.
Theorem 2.13 Let F : D → Rn be C 1 (D) for D ⊂ Rn convex. Moreover,
let x0 ∈ D denote a given starting point for the simplified Newton iteration
(2.72). Assume that the following affine contravariant Lipschitz condition
holds:
80
2 Systems of Equations: Local Newton Methods
F (x) − F (x0 ) v ≤ ωF (x0 )(x − x0 ) · F (x0 )v
(2.73)
for x, x0 ∈ D, v ∈ Rn and 0 ≤ ω < ∞. Define the level set
1 Lω := x ∈ Rn |F (x) ≤
2ω
and let Lω ⊆ D be bounded. Assume that x0 ∈ Lω , which is
h0 := ωF (x0 ) ≤
1
2
.
(2.74)
Then the iterates remain in Lω and converge to a solution point x∗ . The
iterative residual norms converge to zero at an estimated rate
F (xk+1 )
≤ 12 (tk + tk+1 ) < 1 −
F (xk )
1 − 2ho ,
wherein the {tk } are defined by t0 = 0 and
tk+1 = h0 + 12 t2k , k = 0, 1, . . . .
Proof. We apply the Lipschitz condition (2.73) to obtain
F (xk+1 ) =
1 k
k k
F (x + tΔx ) − F (x0 ) Δx dt
t=0
≤
k
ωF (x0 )Δx ·
1
k
F (x0 )(xk − x0 + tΔx )dt
t=0
and, by triangle inequality:
F (xk+1 ) ≤ ωF (xk ) F (x0 )(xk − x0 ) + 12 F (xk ) .
(2.75)
We therefore introduce the majorants
ωF (x0 )(xk − x0 ) ≤ tk
ωF (x0 )(xk+1 − xk ) = ωF (xk ) ≤ hk
with initial values t0 = 0, h0 ≤ 12 . Because of
F (x0 )(xk+1 − x0 ) ≤ F (x0 )(xk − x0 ) + F (x0 )(xk+1 − xk )
and the above relation (2.75), we obtain the same two majorant equations as
in Section 2.1.2
tk+1 = tk + hk , hk = hk−1 tk−1 + 12 hk−1
and from these a single equation of the form
2.2 Residual Based Algorithms
81
tk+1 − tk = (tk − tk−1 ) tk−1 + 12 (tk − tk−1 ) = 12 (t2k − t2k−1 ) .
Rearrangement of this equation permits the application of the Ortega trick
tk+1 − 12 t2k = t1 − 12 t20 = h0 ,
which once again may be interpreted as the simplified Newton iteration
tk+1 − tk = −
g(tk )
= g(tk )
g (t0 )
for the scalar equation
g(t) = h0 − t + 12 t2 = 0 .
As can be seen from the above Figure 2.1, here also we obtain g(tk+1 ) < g(tk ),
which is equivalent to hk+1 < hk and therefore
F (xk+1 ) < F (xk ) ≤
1
.
2ω
This assures that all simplified Newton iterates remain in Lω ⊂ D. As for the
convergence to some (not necessarily unique) solution point x∗ ∈ Lω ⊂ D,
arguments similar to the ones used for Theorem 2.12 can be applied. As for
the convergence rate, we go back to (2.75) and derive
F (xk+1 )
≤ tk + 12 hk = 12 (tk + tk+1 ) < t∗ = 1 −
F (xk )
1 − 2h0 ,
which completes the proof.
Convergence monitor. In order to exploit this theorem for actual implementation, we define the residual contraction factors (k = 0, 1, . . .)
Θk :=
F (xk+1 )
≤ 12 (tk + tk+1 ) .
F (xk )
For k = 0, the local convergence domain is characterized by
Θ0 ≤ 12 h0 ≤
1
4
,
(2.76)
which is clearly more restrictive than the comparable condition Θ0 < 1 for
the ordinary Newton method—compare (2.69).
2.2.3 Broyden’s ‘bad’ rank-1 updates
In this section, we deal with a quasi-Newton update already discussed by
C.G. Broyden in his seminal paper [40] and classified there, on the basis
82
2 Systems of Equations: Local Newton Methods
of his numerical experiments, as being ‘bad’. This method can actually be
derived in terms of affine contravariance. As stated before, only image space
quantities like the residuals Fk := F (xk ) are of interest in this frame. With
δFk+1 = Fk+1 − Fk , we rewrite the secant condition (1.17) here as
Ek (J)δFk+1 = Fk+1
(2.77)
in terms of the affine contravariant update change matrix
Ek (J) := I − Jk J −1 .
Any Jacobian rank-1 update satisfying
Fk+1 vT
−1
−1
Jk+1 = Jk
I− T
, v ∈ Rn , v = 0
v δFk+1
with v some vector in the image space of F will both satisfy the secant
condition and reflect affine contravariance. As an example, the so-called ‘bad’
Broyden method is characterized by setting v = δFk+1 .
Convergence analysis. We start with an analysis of one quasi–Newton
step of this kind.
Theorem 2.14 Let
−1
Jk+1
=
Jk−1
T
Fk+1 δFk+1
I−
δFk+1 2
(2.78)
denote the affine contravariant ‘bad’ Broyden rank-1 update and assume residual contraction
Fk+1 Θk :=
< 1.
Fk Then:
1. The update matrix Jk+1 is a least change update in the sense that
Ek (Jk+1 )
Ek (Jk+1 )
≤ Ek (J)
Θk
≤
.
1 − Θk
∀J ∈ Sk
2. The update matrix Jk+1 is nonsingular whenever Jk is nonsingular and
can be represented by
T
Fk+1 δFk+1
Jk+1 = I −
Jk .
T F
δFk+1
k
2.2 Residual Based Algorithms
83
3. With δxk+1 = −Jk−1 Fk+1 , the next quasi-Newton correction is
T
δFk+1
Fk+1
−1
δxk+1 = −Jk+1 Fk+1 = 1 −
δxk+1 .
δFk+1 2
Proof. For the above rank-1 update we have
Ek (Jk+1 ) =
T
Fk+1 δFk+1
δFk+1 2
and therefore
Ek (Jk+1 ) =
Ek (Jk+1 )δFk+1 Fk+1 Ek (J)δFk+1 =
=
≤ Ek (J)
δFk+1 δFk+1 δFk+1 for all J satisfying the secant condition (2.77). Further, for Θk < 1, we obtain
Ek (Jk+1 ) =
Fk+1 Θk
≤
,
δFk+1 1 − Θk
which confirms the above statement 1. Statements 2 and 3 are direct consequences of the Sherman-Morrison formula.
The above Theorem 2.14 only deals with the situation within one iterative
step. The iteration as a whole is studied next.
Theorem 2.15 For F ∈ C 1 (D), F : D ⊂ Rn → Rn , D convex, let x∗ denote
a unique solution point of F with F (x∗ ) nonsingular. Assume that for some
ω < ∞ the affine contravariant Lipschitz condition
(F (x) − F (x∗ ))(y − x) ≤ ω F (x∗ )(x − x∗ ) F (x∗ )(y − x)
(2.79)
holds for x, y ∈ D. Consider the quasi-Newton iteration as defined in Theorem 2.14. For some Θ in the range 0 < Θ < 1 assume that:
1. in terms of the affine contravariant deterioration matrix
Ek := I − F (x∗ )Jk−1
the initial approximate Jacobian satisfies
η 0 := E0 < Θ ,
2. the initial guess x0 satisfies
t0 := ω F (x∗ )(x0 − x∗ ) ≤
Θ − η0
.
1 + η 0 + 43 (1 − Θ)−1
84
2 Systems of Equations: Local Newton Methods
Then the quasi-Newton iterates xk converge to x∗ in terms of errors as
F (x∗ )(xk+1 − x∗ ) ≤ Θ F (x∗ )(xk − x∗ )
or, in terms of residuals as
Fk+1 ≤ Θ Fk .
The convergence is superlinear with
lim
k→∞
Fk+1 = 0.
Fk (2.80)
As for the Jacobian rank-1 updates, the ‘bounded deterioration property’ holds
in the form
t0
Ek ≤ η 0 +
≤Θ
(1 − t0 )(1 − Θ)
together with the asymptotic property
lim
k→∞
Ek δFk+1 = 0.
δFk+1 Proof. For ease of writing we characterize the Jacobian update approximation by
Ek δFk+1 ηk :=
, ηk := Ek ≥ ηk .
δFk+1 For the convergence analysis we introduce
fk := F (x∗ )(xk − x∗ ) and tk := ω fk .
I. To begin with, we analyze the behavior of the iterative residuals:
1
Fk+1
=
F (xk + sδxk )δxk ds
Fk +
s=0
1
(F (xk + sδxk ) − F (x∗ ))δxk ds + (F (x∗ ) − Jk )δxk .
=
s=0
Applying the Lipschitz condition (2.79) yields
2.2 Residual Based Algorithms
85
1
Fk+1 (F (xk + sδxk ) − F (x∗ ))δxk ds + (F (x∗ )Jk−1 − I)Fk ≤
s=0
1
ωF (x∗ )(xk + sδxk − x∗ ) F (x∗ )δxk ds + Ek Fk ≤
s=0
1
≤
s=0
ω F (x∗ )(1 − s)(xk − x∗ )
+F (x∗ )s(xk+1 − x∗ ) F (x∗ )δxk ds + η k Fk 1
2 (tk
=
+ tk+1 )F (x∗ )δxk + ηk Fk .
Defining t̄k := 12 (tk + tk+1 ), we get
Fk+1 ≤ t̄k (Ek − I)Fk + η k Fk ≤ (t̄k (1 + η k ) + ηk )Fk .
(2.81)
As for the iterative errors fk , we may derive the relation
fk+1
= fk − F (x∗ )Jk−1 Fk = F (x∗ )(xk − x∗ ) − Fk + Ek Fk
1
=
∗
F (x ) − F (x∗ + s(xk − x∗ )) (xk − x∗ ) ds + Ek Fk ,
s=0
from which we obtain the estimate
1
fk+1 sω F (x∗ )(xk − x∗ ) F (x∗ )(xk − x∗ ) ds + η k Fk ≤
s=0
≤
ω
fk 2 + η k (fk − Fk + fk ) .
2
By multiplication with ω and proceeding as above, this can be further reduced
to yield
1 2
1 + ηk
1 2
tk+1 ≤ 2 tk + η k 2 tk + tk = η k +
tk tk .
(2.82)
2
II. Next, we study the approximation properties of the Jacobian updates.
Introducing the orthogonal projection
Qk :=
T
δFk+1 δFk+1
δFk+1 2
onto the secant direction δFk+1 , the deterioration matrix may be written as
Ek+1 = Ek Q⊥
k + Ek+1 Qk ,
(2.83)
86
2 Systems of Equations: Local Newton Methods
yielding, as in the ‘good’ Broyden proof,
η k+1 = Ek+1 ≤ Ek Q⊥
k + Ek+1 Qk ≤ Ek +
Ek+1 δFk+1 .
δFk+1 Using the secant condition (2.77), we get for the numerator of the second
right hand term:
Ek+1 δFk+1
−1
= δFk+1 − F (x∗ )Jk+1
δFk+1 = δFk+1 − F (x∗ )δxk
1
(F (xk + sδxk ) − F (x∗ ))δxk .
=
s=0
This can be estimated as above as follows
≤ t̄k F (x∗ )δxk Ek+1 δFk+1 = t̄k Ek+1 δFk+1 − δFk+1 ≤ t̄k (Ek+1 δFk+1 + δFk+1 )
in order to get
Ek+1 δFk+1 ≤
t̄k
δFk+1 .
1 − t̄k
(2.84)
Inserting this estimate into (2.83) yields the quite rough estimate
η k+1 ≤ η k +
t̄k
.
1 − t̄k
III. For the purpose of repeated induction assume that we have
ηk ≤ η0 +
with
η := η 0 +
k−1
i
Θ t0
≤η
1 − t0
i=0
t0
(1 − t0 )(1 − Θ)
and
k
tk ≤ Θ t0 .
Then by (2.82) and by the subsequent technical Lemma 2.16 below
tk+1 ≤ (η + (1 + η)t0 )tk ≤ Θtk ≤ Θ
k+1
t0
and thus
η k+1
tk
≤ ηk +
≤ η0 +
1 − t0
k−1
i
k+1
Θ t0 Θ
t0
+
≤ η0 +
1 − t0
1 − t0
i=0
k
i
Θ t0
≤ η.
1 − t0
i=0
2.2 Residual Based Algorithms
87
By induction we have the ‘bounded deterioration property’
ηk ≤ η
and the error contraction
tk+1 ≤ tk
for any k. Obviously, by (2.81) and the subsequent technical Lemma 2.16 we
also have contraction of the residuals:
Fk+1 ≤ ΘFk IV. In order to show superlinear convergence, we use the orthogonal splitting
provided by (2.83) in a more subtle manner. Since
Qk EkT v = δFk+1
δFk+1 , EkT v
Ek δFk+1 , v
= δFk+1
,
δFk+1 2
δFk+1 2
some short calculation supplies the equation
T
Ek+1
v2
T
2
T
2
= Q⊥
k Ek v + Qk Ek+1 v
T
= EkT v2 − Qk EkT v2 + Qk Ek+1
v2
= EkT v2 −
Ek δFk+1 , v2
Ek+1 δFk+1 , v2
+
.
2
δFk+1 δFk+1 2
Summing over the indices k, we arrive at
l
k=0
T
El+1
v
Ek δFk+1 , v2
E0T v2
=
−
+
2
2
2
δFk+1 v
v
v2
l
k=0
Ek+1 δFk+1 , v2
.
δFk+1 2 v2
Upon dropping the negative right hand term, letting l → ∞, and using (2.84),
we end up with the estimate
l
k=0
Ek δFk+1 , v2
≤ η20 +
δFk+1 2 v2
l
k=0
tk
1 − tk
2
≤ η 20 +
t20
2
(1 − t0 )2 (1 − Θ )
.
Since the right hand side is bounded, we immediately conclude that
Ek δFk+1 , v2
=0
k→∞ δFk+1 2 v2
lim
for all v ∈ Rn . As a consequence, we must have
lim ηk = 0 .
k→∞
In order to prove the superlinear convergence statement (2.80), we may collect
some estimates from above and proceed as
88
2 Systems of Equations: Local Newton Methods
F (x∗ )Jk−1 Fk+1 = Ek+1 δFk+1 − Ek δFk+1 ≤ t̄k (1 + η k )Fk + ηk δFk+1 ≤ (t̄k (1 + η k ) + ηk (1 + Θ))Fk .
Finally, with
Fk+1 − F (x∗ )Jk−1 Fk+1 ≤
⇒
Fk+1 ≤
Ek Fk+1 ≤ η k Fk+1 F (x∗ )Jk−1 Fk+1 ,
1 − ηk
we get
t̄k (1 + η k ) + ηk (1 + Θ)
Fk .
1 − ηη k
Fk+1 ≤
Since t̄k → 0 and ηk → 0, superlinear convergence is easily verified.
For ease of the above derivation, the following technical lemma has been
postponed.
Lemma 2.16 Assume 0 < Θ < 1, 0 ≤ η0 < Θ and
t≤
Then, with η = η0 +
Θ − η0
.
1 + η0 + 43 (1 − Θ)−1
t
, we have
(1 − t)(1 − Θ)
η + (1 + η)t ≤ Θ .
Proof. Under the given assumptions, a short calculation shows that t < 17 .
Therefore we can proceed as
Θ ≥ η0 + 1 + η0 + 43 (1 − Θ)−1 t
= η0 +
7
6t
1−Θ
+ 1 + η0 + 16 (1 − Θ)−1 t
t
t
≥ η0 +
+ 1 + η0 +
t
(1 − t)(1 − Θ)
(1 − t)(1 − Θ)
= η + (1 + η)t .
2.2 Residual Based Algorithms
89
Algorithmic realization. From representation (2.78) we again have a product form for the Jacobian update inverses. As a condition number monitor
for the possible occurrence of ill-conditioning of the recursive Jacobian rank-1
updates, Lemma 2.8 may once more be applied, here to:
T
Fk+1 δFk+1
cond2 (Jk+1 ) ≤ cond2 I −
cond2 (Jk ).
δFk+1 2
In the present context, we obtain for Θk < 1/2:
cond2 (Jk+1 ) ≤
1
cond2 (Jk ) .
1 − 2Θk
As a consequence, a restriction such as
Θk ≤ Θmax <
1
2
with, say Θmax = 1/4, will be necessary. With these preparations, we are now
ready to present the ‘bad Broyden’ algorithm QNRES (the acronym stands for
RESidual based Quasi-Newton algorithm).
Algorithm QNRES.
F0 := F (x0 )
evaluation and store
σ0 := F0 2
store
J0 δx0 = −F0
linear system solve
κ := 1
For k := 0, 1, . . . , kmax :
xk+1 := xk + δxk
Fk+1 := F (xk+1 )
δFk+1 := Fk+1 − Fk
σk+1 := Fk+1 2
If σk+1 ≤ FTOL2 :
solution found: x∗ = xk+1
Θk :=
σk+1 /σk
If Θk ≥ Θmax :
stop: no convergence
w := δFk+1
γk := w2
κ := κ/(1 − 2Θk )
90
2 Systems of Equations: Local Newton Methods
If κ ≥ κmax :
stop: ill-conditioned update
v := (1 − w, Fk+1 /γk )Fk+1
For j = k − 1, . . . , 0:
β := δFj+1 , v/γj
v = v − βFj+1
J0 δxk+1 = −v
stop: no convergence within kmax iterations
The above algorithm merely requires to store the residuals F0 , . . . , Fk+1 , and
the differences δF1 , . . . , δFk+1 , which means an extra array storage of up
to 2(kmax + 2) vectors of length n. Note that there is a probably machinedependent tradeoff between computation and storage: the vectors δFj+1 can
be either stored or recomputed. Moreover, careful considerations about residual scaling in the inner product ·, · are recommended.
2.2.4 Inexact Newton-RES method
Recall inexact Newton methods with inner and outer iteration formally written as (dropping the inner iteration index i)
F (xk )δxk = −F (xk ) + rk , xk+1 = xk + δxk , k = 0, 1, . . . .
(2.85)
In what follows, we will work out details for GMRES as inner iteration (see
Section 1.4.1). For ease of presentation, we fix the initial values
δxk0 = 0 and r0k = F (xk ) ,
which, during the inner iteration (i = 0, 1, . . .), implies in the generic case
that
rik ηi =
≤ 1 and ηi+1 < ηi , if ηi = 0 .
F (xk )
In what follows, we will denote the final value obtained from the inner iteration in each outer iteration step k by ηk , again dropping the inner iteration
index i.
Convergence analysis. For the inexact Newton-GMRES iteration, we may
state the following convergence theorem.
Theorem 2.17 Let F : D → Rn , F ∈ C 1 (D), D ⊂ Rn convex. Let x0 ∈ D
denote a given starting point for an inexact Newton iteration (2.85). Assume
the affine contravariant Lipschitz condition
2.2 Residual Based Algorithms
91
F (y) − F (x) (y − x) ≤ ωF (x)(y − x)2
for 0 ≤ ω < ∞ , and x, y ∈ D .
Let the level set L0 := x ∈ Rn | F (x) ≤ F (x0 ) ⊆ D be compact. For
each well-defined iterate xk ∈ D define hk := ωF (xk ). Then the outer
residual norms can be bounded as
F (xk+1 ) ≤ ηk + 12 (1 − ηk2 )hk F (xk ) .
(2.86)
The convergence rate can be estimated as follows:
I. Linear convergence mode. Assume that the initial guess xo gives rise
to
h0 < 2 .
Then some Θ in the range h0 /2 < Θ < 1 can be chosen. Let the inner GMRES
iteration be controlled such that
ηk ≤ Θ − 12 hk .
(2.87)
Then the Newton-GMRES iterates {xk } converge at least linearly to some solution point x∗ ∈ L0 at an estimated rate
F (xk+1 ) ≤ ΘF (xk ) .
II. Quadratic convergence mode. If, for some ρ > 0, the initial guess x0
guarantees that
h0 < 2/(1 + ρ)
and the inner iteration is controlled such that
ηk
≤ 12 ρhk ,
1 − ηk2
(2.88)
then the convergence is quadratic at an estimated rate
F (xk+1 ) ≤ 12 ω(1 + ρ)(1 − ηk2 )F (xk )2 .
(2.89)
Proof. Proceeding as in earlier proofs, we obtain
F (xk+1 )
= 1 F (xk + tδxk ) − F (xk ) δxk dt + rk 0
1 k
F (x + tδxk ) − F (xk ) δxk dt + rk ≤
≤
0
1
ωF (xk )
2
− rk 2 + rk .
By use of (1.20), this is seen to be just (2.86). Under the assumption (2.87)
with Θ < 1 and ηk < 1 from GMRES, we obtain
92
2 Systems of Equations: Local Newton Methods
F (xk+1 ) ≤ ΘF (xk )
and by repeated induction
{xk } ⊂ L0 ⊂ D ,
from which the convergence to x∗ ∈ L0 is concluded. Quadratic convergence
as in (2.89) is shown by mere insertion of (2.88) into (2.86).
Convergence monitor. Throughout the inexact Newton iteration we will
check for residual monotonicity
Θk :=
F (xk+1 )
≤ Θ < 1 , k = 0, 1, . . . ,
F (xk )
introducing certain default parameters Θ in accordance with the above Theorem 2.17. We will regard an iteration as divergent, whenever Θk ≥ Θ holds.
Termination criterion. As in the exact Newton iteration, the finally accepted iterate x̂ is required to satisfy
F (x̂) ≤ FTOL
with FTOL a user prescribed residual error tolerance.
Standard convergence mode. If ηk ≤ η̄ < 1 is prescribed by the user,
then (2.86) implies that Θk → η̄ and asymptotic linear convergence occurs—
as already shown in the early pioneering paper [51].
Quadratic convergence mode. Assume that for k = 0 some value η0 is
prescribed; from numerical experiments, we know that this value should be
sufficiently small—compare, e.g., Table 8.3 in Section 8.2 below. For k ≥ 0,
(2.89) suggests the a-posteriori estimate
[hk ]2 :=
2Θk
≤ hk
(1 + ρ)(1 − ηk2 )
and, since hk+1 = Θk hk , also the a-priori estimate:
[hk+1 ] := Θk [hk ]2 ≤ hk+1 .
For k > 0, shifting the index k + 1 now back to k, we therefore require that
ηk
≤ 12 ρ[hk ] ≤ 12 ρhk ,
1 − ηk2
(2.90)
which can be assured in the course of the iterative computation of δxk and
rk . For the parameter ρ some value ρ ≈ 1 seems to be appropriate. Note that
asymptotically this choice leads to ηk → ρ[hk ] → 0.
2.2 Residual Based Algorithms
93
Linear convergence mode. Once the local contraction factor Θk is sufficiently below some prescribed value Θ, we may switch to the linear convergence mode described in the above Theorem 2.17. Careful examination of the
proof shows that
F (xk+1 ) − rk ≤
ω
F (xk ) − rk 2 = 12 (1 − ηk2 )hk F (xk ) .
2
From this we may derive the a-posteriori estimate
[hk ]1 :=
2F (xk+1 ) − rk ≤ hk
(1 − ηk2 )F (xk )
and, since hk+1 = Θk hk , also the a-priori estimate
[hk+1 ] := Θk [hk ]1 ≤ hk+1 .
As a preparation of the next Newton step, we define
η k+1 = Θ − 12 [hk+1 ]
in terms of the above a-priori estimate. If this value is smaller than the
value obtained from (2.90), then we continue the iteration in the quadratic
convergence mode. Else, we realize the linear convergence mode in Newton
step k + 1 with some
ηk+1 ≤ η k+1 .
Asymptotically, this strategy leads to ηk+1 → Θ.
Preconditioning. In order to speed up the inner iteration, preconditioning
from the left or/and from the right may be used. This means solving
−1 k CL F (xk )CR CR
δx = CL −F (xk ) + rk
instead of (2.85). In such a case, the norm of the preconditioned residuals r̄k =
CL rk is minimized in GMRES, whereas CR only affects the rate of convergence
via the Krylov subspace
Ki (r̄0 , A) with A = CL F (xk )CR .
Consequently, the above strategy should be based on the contraction factors
Θk =
CL F (xk+1 )2
CL F (xk )2
for the outer iteration. Note, however, that CL should not depend on the
iterate xk in this theoretical setting.
If strict residual minimization is aimed at, then only right preconditioning
should be implemented (i.e., CL = I).
94
2 Systems of Equations: Local Newton Methods
The here described local Newton-GMRES algorithm is part of the global Newton code GIANT-GMRES, which will be described in Section 3.2.3 below.
Remark 2.2 If GMRES were replaced by some other residual norm reducing
(but not minimizing) iterative linear solver, then a similar accuracy matching
strategy can be worked out (left as Exercise 2.9).
Bibliographical Note. The concept of local inexact Newton methods—
sometimes also called truncated Newton methods—seems to have first been
published in 1982 by R.S. Dembo, S.C. Eisenstat, and T. Steihaug [51]; they
presented an asymptotic analysis in terms of the residuals. In 1981, R.E. Bank
and D.J. Rose [19] worked out details of an inexact Newton algorithm on the
basis of residual control including certain algorithmic heuristics. In 1996,
S.C. Eisenstat and H.F. Walker [91] suggested a further strategy to choose
the ηk , which they call ‘forcing terms’; their strategy is also based on convergence analysis results, but different from the one presented here.
2.3 Convex Optimization
In this section we consider the problem of minimizing a strictly convex functional f : D ⊂ Rn −→ R1 . Then F (x) = f (x)T is a gradient mapping and
F (x) = f (x) is symmetric positive definite. We want to solve F (x) = 0, a
system of n nonlinear equations, by local Newton methods. The convergence
analysis will start from affine conjugate Lipschitz conditions of the type (1.9)
and lead to results in terms of iterative functional values and energy norms
of corrections or errors.
2.3.1 Ordinary Newton method
Recall the ordinary Newton method in the notation (k = 0, 1, . . .)
F (xk )Δxk = −F (xk ) , xk+1 = xk + Δxk .
Convergence analysis. We analyze its convergence behavior in terms of
iterative values of the functional to be minimized and energy norms of the
Newton corrections. Thus we arrive at an affine conjugate variant of the
Newton-Mysovskikh theorem.
Theorem 2.18 Let f : D → R1 be a strictly convex C 2 -functional to be
minimized over some open and convex domain D ⊂ Rn . Let F (x) = f (x)T
and F (x) = f (x), which is symmetric and assumed to be strictly positive
definite. Assume that the following affine conjugate Lipschitz condition holds:
2.3 Convex Optimization
−1/2 F (z)
F (y) − F (x) (y − x) ≤ ωF (x)1/2 (y − x)2
95
(2.91)
for collinear x, y, z ∈ D with 0 ≤ ω < ∞. For the initial guess x0 assume
that
h0 = ωF (x0 )1/2 Δx0 < 2
(2.92)
and that the level set L0 := {x ∈ D |f (x) ≤ f (x0 )} is compact. Then the
ordinary Newton iterates remain in L0 and converge to the minimum point
x∗ at a rate estimated by
F (xk+1 )1/2 Δxk+1 ≤ 12 ωF (xk )1/2 Δxk 2
(2.93)
or, with k := F (xk )1/2 Δxk 2 and hk := ωF (xk )1/2 Δxk , by
− 16 hk k
1
6 k
≤ f (xk ) − f (xk+1 ) − 12 k
≤
1
h 6 k k
≤
≤
5
6 k
f (xk ) − f (xk+1 )
(2.94)
.
The distance to the minimum can be bounded as
f (x0 ) − f (x∗ ) ≤
5
6 0
1 − h0 /2
.
Proof. With the Lipschitz condition (2.91) for z = xk+1 , y = xk + tΔxk ,
x = xk , the result (2.93), which is equivalent to hk+1 ≤ h2k /2, is proven just
as before in Theorem 2.2. The fact that xk+1 ∈ L0 can be seen by applying
the same technique as in the proof of Theorem 2.12 above. To derive (2.94),
we verify that
1
1 k f (xk+1 ) − f (xk ) + 12 F (xk )1/2 Δxk 2 =
s
Δx , w dtds ,
s=0 t=0
k
where w = F (x + stΔxk ) − F (xk ) Δxk
(2.95)
with ·, · the Euclidean inner product. The integrand term is estimated as
Δxk , w ≤
≤
| F (xk )1/2 Δxk , F (xk )−1/2 w |
F (xk )1/2 Δxk · ωstF (xk )1/2 Δxk 2
by the Cauchy-Schwarz inequality and (2.91) with x = z = xk , y = xk +
stΔxk . With hk < 2 this is the left side of (2.94). Consequently, the iterates
converge to x∗ . Note that x∗ is anyway unique in D under the assumptions
made.
In order to obtain the right hand side of (2.94), we go up to (2.95), but this
time apply Cauchy-Schwarz in the other direction, which yields:
0 ≤ f (xk ) − f (xk+1 ) ≤ 12 + 16 hk F (xk )1/2 Δxk 2 < 56 k .
96
2 Systems of Equations: Local Newton Methods
Summing over all k = 0, 1, . . . we get
∞
0 ≤ ω 2 f (x0 ) − f (x∗ ) ≤
1
2
1 3
2 hk + 6 hk <
k=0
By using
1
2 hk+1
≤
1
2 hk
∞
h2k .
5
6
k=0
2
≤ 12 hk < 1
the right hand upper bound can be further treated to obtain
( 12 h0 )2 + ( 12 h1 )2 + · · ·
≤ ( 12 h0 )2 + ( 12 h0 )4 + ( 12 h1 )4 + · · ·
∞
1 2
4 h0
<
( 12 h0 )k =
k=0
so that
ω 2 f (x0 ) − f (x∗ ) <
1
5 2
h
6 0
− 12 h0
1
1 2
4 h0
− 12 h0
,
.
This is the last statement of the theorem.
Convergence monitor. We now study the consequences of the above convergence theorem for actual implementation. Let k , Θk be defined as
1/2
k+1
k = F (xk )1/2 Δxk 22 = | F (xk ), Δxk | , Θk =
.
k
Then the basic convergence result is
Θk =
hk+1
≤ 12 hk < 1
hk
and
f (xk+1 ) − f (xk ) < − 16 k .
For k = 0, we must have
Θ0 < 1
0
to assure that x is within the local convergence domain. For k > 0, in a
similar way as in the two cases before, we derive the approximate recursion
(k = 0, 1, . . .)
Θk+1 ≈ Θk2 < Θ0 < 1 .
From this, we may terminate the iteration as ‘divergent’ whenever
f (xk+1 ) − f (xk ) ≥ − 16 k
or, since this criterion is prone to suffer from rounding errors, either
Θk ≥ Θ0
or
Θk+1 ≥
(k > 0),
Θk2
.
Θ0
2.3 Convex Optimization
97
Termination criterion. We may terminate the iteration whenever either
k ≤ ETOL2
or, recalling that asymptotically
.
f (xk+1 ) − f (xk ) = − 12 k ,
whenever
f (xk ) − f (xk+1 ) ≤
1
2
ETOL2
with ETOL a user prescribed energy error tolerance.
2.3.2 Simplified Newton method
Recall the notation of the simplified Newton iteration
k
k
F (x0 )Δx = −F (xk ) , xk+1 = xk + Δx , k = 0, 1, . . . .
Convergence analysis. We now want to study its functional minimization properties, when the Jacobian matrix is kept throughout the Newton
iteration.
Theorem 2.19 Let f : D → R1 be a strictly convex C 2 -functional to be
minimized over some convex domain D ⊂ Rn . Let F (x) = f (x)T and
F (x) = f (x), which is then symmetric positive definite. Let x0 ∈ D be
some given starting point for a simplified Newton iteration. Assume that the
following affine conjugate Lipschitz condition holds:
F (x0 )−1/2 (F (z) − F (x0 ))v ≤ ωF (x0 )1/2 (z − x0 ) · F (x0 )1/2 v
for z ∈ D. Let
0
h0 := ωF (x0 )1/2 Δx ≤
1
2
√
k
and define t∗ = 1 − 1 − 2h0 . Then, with k := F (x0 )1/2 Δx 2 , the simplified Newton iteration converges to some x∗ with
ωx∗ − x0 ≤ t∗ .
The convergence rate can be estimated in terms of the functional by
− 16 k (tk+1 + 2tk ) ≤ f (xk ) − f (xk+1 ) − 12 k ≤ 16 k (tk+1 + 2tk )
or in terms of energy norms of the simplified Newton corrections by
1/2
k+1
Θk =
≤ 12 (tk+1 + tk ) ,
k
wherein {tk } is defined from t0 = 0 and
tk+1 = h0 + 12 t2k < t∗ , k = 0, 1, . . . .
(2.96)
98
2 Systems of Equations: Local Newton Methods
Proof. The proof is similar to the previous proofs of Theorem 2.5 and Theorem 2.13 and will therefore only be sketched here. With the definition for
k and the majorants
k
ωF (x0 )1/2 (xk − x0 ) ≤ tk , ωF (x0 )1/2 Δx ≤ hk
we obtain for the functional decrease
f (xk+1 ) − f (xk ) + 12 k =
1
1 k k
k
k
=
s
Δx , F (x + tsΔx ) − F (x0 ) Δx dtds
s=0
≤ ωk
t=0
1
s=0
s
1 (1 − ts)F (x0 )1/2 (xk − x0 ) +
t=0
+ tsF (x0 )1/2 (xk+1 − x0 ) dtds
≤ 16 k (tk+1 + 2tk ) .
This is the basis for (2.96). The energy norm contraction factor arises as
1/2
k+1
hk+1
Θk =
≤ 12 (tk + tk+1 ) =:
.
k
hk
With t0 = 0, tk+1 = tk + hk and the usual ‘Ortega trick’ the results above
are essentially established.
Convergence monitor. For actual computation, we also have
Θ0 ≤ 12 h0 ≤
1
4
.
Note that for the simplified Newton iteration, the asymptotic property
f (x∗ ) − f (xk ) ≈ 12 k does not hold—compare (2.96). Mutatis mutandis, essentially just replacing norms by energy norms in the contraction factors Θk ,
the techniques already worked out in Section 2.1.2 carry over.
Termination criterion. This also can be directly copied from Section 2.1.2
with the proper replacement of norms by energy norms.
2.3.3 Inexact Newton-PCG method
We next study inexact Newton methods (dropping, as usual, the inner iteration index i)
F (xk )(δxk − Δxk ) = rk , xk+1 = xk + δxk ,
k = 0, 1, . . . .
(2.97)
In the context of (strictly) convex optimization the Jacobian matrices can be
assumed to be symmetric positive definite, so that the outstanding candidate
for an inner iteration will be the preconditioned conjugate gradient (PCG).
Throughout this section we set δxk0 = 0.
2.3 Convex Optimization
99
Convergence analysis. For the purpose of our analysis below, we recall
the following orthogonality condition, which is equivalent to condition (1.21)
independent of the selected preconditioner:
δxk , F (xk )(δxk − Δxk ) = δxk , rk = 0 .
(2.98)
As before, Δxk denotes the associated exact Newton correction. After these
preparations, we are now ready to derive a Newton-Mysovskikh type theorem,
which meets our above affine conjugacy requirements.
Theorem 2.20 Let f : D → R be a strictly convex C 2 -functional to be
minimized over some open and convex domain D ⊂ Rn . Let F (x) := f (x)
be symmetric positive definite and let · denote the Euclidean vector norm.
In the above introduced notation assume the existence of some ω < ∞ such
that the following affine conjugate Lipschitz condition holds for collinear x,
y, z ∈ D:
−1/2 F (z)
F (y) − F (x) v ≤ ω F (x)1/2 (y − x) · F (x)1/2 v .
Consider an inexact Newton-PCG iteration (2.97) satisfying (2.98) and started
with δxk0 = 0. At any well-defined iterate xk , define the exact Newton terms
k := F (xk )1/2 Δxk 2 and hk := ω F (xk )1/2 Δxk and, subject to inner iteration errors characterized by
δk :=
F (xk )1/2 (δxk − Δxk )
,
F (xk )1/2 δxk the associated inexact Newton terms
δk := F (xk )1/2 δxk 2 =
k
and hδk := ω F (xk )1/2 δxk =
1 + δk2
hk
.
1 + δk2
For a given initial guess x0 ∈ D assume that the level set L0 :=
{x ∈ D | f (x) ≤ f (x0 )} is closed and bounded. Then the following results
hold:
I. Linear convergence mode. Assume that x0 satisfies
h0 < 2Θ < 2
(2.99)
for some Θ < 1. Let δk+1 ≥ δk throughout the inexact Newton iteration.
Moreover, let the inner iteration be controlled such that
hδk + δk hδk + 4 + (hδk )2
ϑ(hδk , δk ) :=
≤ Θ,
(2.100)
2 1 + δk2
100
2 Systems of Equations: Local Newton Methods
which assures that
δk ≤ Θ/
2
1−Θ .
(2.101)
Then the iterates xk remain in L0 and converge at least linearly to the minimum point x∗ ∈ L0 such that
F (xk+1 )1/2 Δxk+1 ≤ Θ F (xk )1/2 Δxk and
(2.102)
F (xk+1 )1/2 δxk+1 ≤ Θ F (xk )1/2 δxk .
II. Quadratic convergence mode. Let for some ρ > 0 the initial iterate
x0 satisfy
2
hδ0 <
(2.103)
1+ρ
and the inner iteration be controlled such that
δk ≤
ρhδ
k
,
hδk + 4 + (hδk )2
which requires that
δ0 <
ρ
1+
1 + (1 + ρ)2
.
(2.104)
(2.105)
Then the inexact Newton iterates xk remain in L0 and converge quadratically
to the minimum point x∗ ∈ L0 such that
and
ω
F (xk+1 )1/2 Δxk+1 ≤ (1 + ρ) F (xk )1/2 Δx2
2
(2.106)
ω
F (xk+1 )1/2 δxk+1 ≤ (1 + ρ) F (xk )1/2 δx2 .
2
(2.107)
III. Functional descent. The convergence in terms of the functional can
be estimated by
− 61 hδk δk ≤ f (xk ) − f (xk+1 ) − 12 δk ≤ 16 hδk δk .
(2.108)
Proof. For the purpose of repeated induction, let Lk denote the level set
defined in analogy to L0 . First, in order to show that xk+1 ∈ Lk , we start
from the identity
f (xk + λδxk ) − f (xk ) + λ − 12 λ2 δk
λ
=
λ
s
s=0
t=0
k
δx , (F (xk + stδxk ) − F (xk )) δxk dt ds + δxk , rk .
2.3 Convex Optimization
101
The second right hand term vanishes due to (2.98). The energy product in
the first term can be bounded as
δxk , . . . ≤ F (xk )1/2 δxk ωstF (xk )1/2 δxk 2 = sthδk δk .
For the purpose of repeated induction, let hk < 2 and k = 0, which then
implies that
f (xk + λδxk ) ≤ f (xk ) + 13 λ3 + 12 λ2 − λ δk < f (xk ) for λ ∈ ]0, 1] .
Therefore, the assumption xk + δxk ∈
/ Lk would lead to a contradiction for
some λ ∈ ]0, 1].
For λ = 1, we get the left hand side of (2.108). Applying the Cauchy-Schwarz
inequality in the other direction also yields the right hand side.
In order to monitor the behavior of the Kantorovich type quantities hk , we
estimate the local energy norms as
F (xk+1 )1/2 Δxk+1 1
k+1 −1/2
k
k
k
k
k ≤
F
(x
)
F
(x
+
tδx
)
−
F
(x
)
δx
dt
+
r
t=0
≤ 12 ω F (xk )1/2 δxk 2 + F (xk+1 )−1/2 rk .
With z = δxk − Δxk , the second right hand term can be estimated implicitly
by
F (xk+1 )−1/2 rk 2 ≤ F (xk )1/2 z2 + hδk F (xk )1/2 z F (xk+1 )−1/2 rk ,
which leads to the explicit bound
2
F (xk+1 )−1/2 rk ≤ 12 hδk + 4 + hδk
F (xk )1/2 z .
Summarizing, we obtain the contraction factor bound
Θk :=
F (xk+1 )1/2 Δxk+1 ≤ ϑ(hδk , δk ) .
F (xk )1/2 Δxk (2.109)
Herein linear convergence shows up via (2.100) and (2.102). The result (2.101)
is obtained with hk = 0. Obviously, hk < 2Θ is necessary to obtain Θk ≤ Θ
for some Θ < 1. As for the contraction of the inexact corrections, we apply
δk+1 ≥ δk and (1.26) to show that
1 + δk2
F (xk+1 )1/2 δxk+1 =
Θk ≤ Θk ≤ Θ .
2
1 + δk+1
F (xk )1/2 δxk Hence, we may complete the induction and conclude that the iterates xk
converge to x∗ .
102
2 Systems of Equations: Local Newton Methods
As for quadratic convergence, we impose condition (2.104) within (2.109) to
obtain
F (xk+1 )1/2 Δxk+1 1
δ
δ
δ )2
≤
h
+
δ
(h
+
4
+
(h
k k
k
k
F (xk )1/2 Δxk 2 1 + δk2
≤
1
(1
2
+ ρ)hδk ,
which, for hδk ≤ hk ≤ h0 assures the convergence relations (2.106) under the
assumption (2.99). Upon inserting (2.103) into (2.104) we immediately verify
(2.105). For the inexact corrections, we have equivalently
F (xk+1 )1/2 δxk+1 F (xk )1/2 δxk ≤
≤
1
2
2 1 + δk+1
1
(1
2
hδk + δk (hδk + 4 + (hδk )2
+ ρ)hδk < 1 ,
which then assures the convergence relations (2.107). This finally completes
the proof.
Convergence monitor. Assume now that we have a reasonable (and cheap)
estimate of the relative energy norm errors δk available from the inner PCG
iteration. A new iterate xk+1 might be accepted whenever either
f (xk+1 ) − f (xk ) ≤ − 61 k = − 61 (1 + δk2 )δk .
or, as a slight generalization of the situation of Theorem 2.20, the inexact
monotonicity criterion
Θk :=
k+1
k
1/2
=
2
(1 + δk+1
)δk+1
2 δ
(1 + δk )k
1/2
≤ Θk < 1
holds. We will regard the outer iteration as divergent, if none of the above
criteria is met.
Termination criteria. We will terminate the iteration whenever
k = (1 + δk2 )δk ≤ ETOL2
or f (xk ) − f (xk+1 ) ≤
1
2
ETOL2 .
(2.110)
Standard convergence mode. If we just impose the inner iteration termination criterion δk ≤ δ̄ for some fixed default value δ̄, we obtain
√ asymptotic
linear convergence. If we set Θ = 12 , then (2.101) induces δ̄ < 3/3. As in the
other two cases, we recommend δ̄ = 1/4 to assure at least two binary digits.
2.3 Convex Optimization
103
Quadratic convergence mode. Assume that [h0 ] < 2/(1 + ρ) for ρ = 1.
Let δ0 be given, say δ0 = 1/4 in agreement with (2.105). As for the adaptive
termination of the inner iteration within the inexact local Newton method,
we want to satisfy condition (2.104). Following our general paradigm, we will
replace the unavailable upper bound therein by the computationally available
condition in terms of computational estimates [hk ] such that
δk ≤
ρ [hδk ]
.
[hδk ] + 4 + [hδk ]2
(2.111)
Since the above right hand side is a monotone increasing function of [hk ], the
relation [hk ] ≤ hk implies that the theoretical condition (2.104) is actually assured whenever (2.111) holds. Following our basic paradigm (compare Section
1.2), we apply (2.108) and define the computational a-posteriori estimates
6
[hδk ]2 = δ |f (xk+1 ) − f (xk ) + 12 δk | , [hk ]2 = 1 + δk2 [hδk ]2 .
k
From this, shifting the index k + 1 back to k, we may define the a-priori
estimate
[hk ] = Θk−1 [hk−1 ]2 ,
(2.112)
which we insert into (2.111) to obtain a simple implicit scalar equation for
δk .
Note that δk → 0 is forced when k → ∞. In words: the closer the iterates come
to the solution point, the more work needs to be done in the inner iteration
to assure quadratic convergence of the outer iteration.
Linear convergence mode. Once the local contraction factor Θk is sufficiently below some prescribed value Θ, we may switch to the linear convergence mode described by the above Theorem 2.20. As for the termination of
the inner iteration, we would like to assure condition (2.100), briefly recalled
as
ϑ(hδk , δk ) ≤ Θ .
Since the above quantity ϑ is unavailable, we will replace it by the computationally available estimate
[ϑ(hδk , δk )] := ϑ([hδk ], δk ) ≤ ϑ(hδk , δk ) .
For k > 0, we may again insert the a-priori estimate (2.112) above. In any
case, we will run the inner iteration until the actual δk satisfies either condition (2.100) for the linear convergence mode or condition (2.111) for the
quadratic convergence mode. Whenever Θk ≥ 1 occurs, then we switch to
some global variant of this local inexact Newton method—see Section 3.4.3.
104
2 Systems of Equations: Local Newton Methods
Note that asymptotically
2
δk → Θ/ 1 − Θ
as
k → ∞.
(2.113)
In other words: the closer the iterates come to the solution point, the less
work is necessary within the inner iteration to assure linear convergence of
the outer iteration.
The here described local inexact Newton algorithm for convex optimization
is part of the global inexact Newton code GIANT-PCG worked out in detail in
Section 3.4.3 below.
Bibliographical Note. The presentation in this chapter is a finite dimensional restriction of the affine conjugate convergence theory and the corresponding algorithmic concepts given by P. Deuflhard and M. Weiser [84]
for nonlinear elliptic PDEs. Our here developed inexact Newton-PCG algorithm may be regarded as a competitor to nonlinear CG methods—both to
the variant [93] due to R. Fletcher and C.M. Reeves and to the one due to
E. Polak and R. Ribière [169, Section 2.3]. For the application of nonlinear
CG to discrete partial differential equations see, e.g., the lecture notes [102] by
R. Glowinski; from this perspective, our Newton-PCG method may be viewed
as a nonlinear CG variant with Jacobian savings in a firm theoretical frame.
Exercises
Exercise 2.1 Derive the computational complexity bounds (2.71) in terms
of number of iterations from Theorem 2.12.
Exercise 2.2 Let M (x) denote a perturbed Jacobian matrix of the form
M (xk ) = F (xk ) + δM (xk ). Derive a convergence theorem for a Newton-like
method based on Theorem 2.10.
Exercise 2.3 As an illustration of the not affine covariant classical NewtonMysovskikh theorem take X = Y = R2 and define
x1 − x2
F (x) :=
.
(x1 − 8)x2
Verify that here hF = αF βF γF < 2. The simple affine transformation
1 1
F → G :=
F
0 12
induces the associated quantities αG , βG , γG , hG . Once more, give best possible bounds and verify that now hG > 2! Finally, prove that the affine invariant
characterization from Theorem 2.2 yields h0 = αω 2. Interpretation?
Exercises
105
Hint: One obtains hF = 0.762, hG = 2.159, h0 = 0.127.
Exercise 2.4 Theorem of H.B. Keller. Let F : D → Rn be a continuously
differentiable mapping with D ⊂ Rn convex. Suppose that F (x) is invertible
for each x ∈ D and satisfies the affine invariant Hölder continuity
−1 F (z)
F (y) − F (x) ≤ ωy − xγ ,
where 0 < γ ≤ 1.
a) Prove a variant of the affine covariant Newton-Mysovskikh theorem
(Theorem 2.2).
b) Prove a variant of the affine covariant Newton-Kantorovich theorem
(Theorem 2.1).
Exercise 2.5 Theorem of L.B. Rall (improved by W.C. Rheinboldt). Let
F : D ⊆ Rn → Rn , D open convex. Assume that there exists a unique
solution x∗ ∈ D and that F (x∗ ) is invertible. Let
∗ −1 F (x )
F (y) − F (x) ≤ ω∗ y − x for x , y ∈ D
denote a special affine covariant Lipschitz condition. Let
S(x∗ , ρ) := {x ∈ X| x − x∗ < ρ} ⊂ D .
By introduction of the majorants
ω∗ xk − x∗ ≤ tk
2
, the ordinary
3ω∗
∗
Newton iteration remains in S and converges to x . Give a convergence rate
estimate.
prove that for any starting point x0 ∈ S(x∗ , ρ) with ρ :=
Exercise 2.6 For convex optimization there are three popular symmetric
Jacobian rank-2 updates
• Broyden-Fletcher-Goldfarb-Shanno (BFGS):
Jk+1 = Jk −
Fk FkT
(Fk+1 − Fk )(Fk+1 − Fk )T
+
,
(Fk+1 − Fk )T δxk
δxTk Jk δxk
• Davidon-Fletcher-Powell (DFP):
Jk+1
= Jk +
−
T
Fk+1 (Fk+1 − Fk )T + (Fk+1 − Fk )Fk+1
−
(Fk+1 − Fk )T δxk
T
Fk+1
δxk
(Fk+1 − Fk )(Fk+1 − Fk )T ,
((Fk+1 − Fk )T δxk )2
106
2 Systems of Equations: Local Newton Methods
• Powell’s symmetric Broyden (PSB):
Jk+1 = Jk +
T
T
Fk+1 δxTk + δxk Fk+1
Fk+1
δxk
−
δxk δxTk .
T
T
δxk δxk
(δxk δxk )2
a) Show that all updates satisfy the classical secant condition.
b) Which of these updates are defined in an affine conjugate way? For not
affine conjugate updates: design an appropriate scaling so that at least
scaling invariance is achieved.
c) Which of these updates can be interpreted as a least change secant update? Derive the associated error concept.
Exercise 2.7 Rank-2 update formulas for convex optimization. We consider
several update formulas for convex optimization. Common basis for all these
updates is the classical secant condition
Jδxk = F (xk + δxk ) − F (xk ) = Fk+1 − Fk = δFk+1 .
a) Show that u and v in the general symmetric positive definite update
formula
J = (I − uv T ) Jk (I − vuT )
cannot be specified such that both the secant condition is satisfied and
the update is of full rank 2.
b) Verify that this can be achieved by the comparable representation, the
DFP update:
T
T
δxk δFk+1
δFk+1 δFk+1
δFk+1 δxTk
Jk+1 = I −
J
I
−
+
.
k
T δx )
T δx )
T δx )
(δFk+1
(δFk+1
(δFk+1
k
k
k
c) Verify that this can be also achieved by the inverse representation, the
BFGS update:
T
T
δx
δF
δF
δx
δxδxTk
k
k+1
k+1
−1
−1
k
Jk+1
= I−
J
I
−
+
.
k
T
T
T
(δFk+1
δxk )
(δFk+1
δxk )
δFk+1
δxk
Exercise 2.8 Recall the notation for quasi-Newton methods as given in
Section 2.1.4. With the majorant definitions
Δxk+1 ≤ Θk < 12 , Δxk ≤ ek ,
Δxk −1 k
J
F (x ) − Jk ≤ δk ,
k
−1 J [F (u) − F (v)] ≤
k
ωk u − v ,
Exercises
107
verify the following set of recursions:
δk+1
ek+1
ωk+1
Θk+1
[δk + Θk + ωk ek ]
1 − Θk ,
Θk
=
ek ,
1 − Θk
ωk
=
,
1 − Θk
= δk+1 + 12 ωk+1 ek+1 .
=
Under the additional assumption of ‘bounded deterioration’ in the form
δk ≤ δ
derive a Kantorovich-type local convergence theorem. Why is such a theorem
unsatisfactory?
Exercise 2.9 Consider a residual based inexact Newton method, where the
inner iteration is done by some residual norm reducing, but not minimizing,
iterative solver—like the ‘bad’ Broyden algorithm BB for linear systems as
described in [74]. Then the contraction results (2.86), which hold for the
residual minimizer GMRES, must be replaced.
a) Show the alternative contraction result
Θk ≤ ηk + 12 (1 + ηk )2 hk .
b) For the Kantorovich quantities hk , find cheap and reliable a-posteriori
and a-priori computational estimates [hk ] ≤ hk .
c) Design accuracy matching strategies (standard, linear, and quadratic convergence mode) similar to those worked out for GMRES in Section 2.2.4.
Exercise 2.10 Consider two Newton sequences {xk }, {y k } starting at different initial guesses x0 , y 0 and continuing as
xk+1 = xk + Δxk ,
y k+1 = y k + Δy k ,
where Δxk , Δy k are the corresponding ordinary Newton corrections. Upon
using the affine covariant Lipschitz condition
F (u)−1 (F (v) − F (w)) u ≤ ωv − wu
verify the nonlinear perturbation result
xk+1 − y k+1 ≤ ω 12 xk − y k + Δxk xk − y k .
Is the result invariant under x ↔ y?
3 Systems of Equations: Global Newton
Methods
As in the preceding chapter, the discussion here is also restricted to systems
of n nonlinear equations, say
F (x) = 0 ,
where F ∈ C 1 (D), D ⊆ Rn , F : D −→ Rn with Jacobian (n, n)-matrix F (x).
In contrast to the preceding chapter, however, available initial guesses x0 of
the solution point x∗ are no longer assumed to be ‘sufficiently close’ to x∗ .
In order to specify the colloquial term ‘sufficiently close’, we recur to any of
the local convergence conditions of the preceding chapter. Let ω denote an
affine covariant Lipschitz constant. Then Theorem 2.3 presents an appropriate local convergence condition of the form
x∗ − x0 < 2/ω .
In the error oriented framework, Theorem 2.2 yields a characterization in
terms of the Kantorovich quantity
h0 := Δx0 ω < 2 ,
which restricts the ordinary Newton correction Δx0 . Under any of these conditions local Newton methods are guaranteed to converge. Such problems
are sometimes called mildly nonlinear. Their computational complexity is a
priori bounded in terms of the computational complexity of solving linear
problems of the same structure—see, for example, the bound (2.71).
In contrast to that, under a condition of the type h0 1, which is equivalent
to
Δx0 2/ω
(3.1)
local Newton methods will not exhibit guaranteed convergence. In this situation, the computational complexity cannot be bounded a priori. Such problems are often called highly nonlinear. Nevertheless, local Newton methods
may actually converge for some of these problems even in the situation of
condition (3.1). A guaranteed convergence, however, will only occur, if additional global structure on F can be exploited: as an example, we treat convex
nonlinear mappings in Section 3.1.1 below.
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance
and Adaptive Algorithms, Springer Series in Computational Mathematics 35,
DOI 10.1007/978-3-642-23899-4_3, © Springer-Verlag Berlin Heidelberg 2011
109
110
3 Systems of Equations: Global Newton Methods
For general mapping F , a globalization of local Newton methods must be
constructed. In Section 3.1 we survey globalization concepts such as
•
•
•
•
steepest descent methods,
trust region methods,
the Levenberg-Marquardt method, and
the Newton method with damping strategy.
In Section 3.1.4, a rather general geometric approach is taken: the idea is to
derive a globalization concept without pre-occupation to any of the known
iterative methods, just starting from the requirement of affine covariance as
a ‘first principle’. Surprisingly, this general approach leads to the derivation
of Newton’s method with damping strategy.
Monotonicity tests. Monotonicity tests serve the purpose to accept or
reject a new iterate. We study different such tests, according to different
affine invariance requirements:
• the most popular residual monotonicity test, which is based on affine contravariance (Section 3.2),
• the error oriented so-called natural monotonicity test, which is based on
affine covariance (Section 3.3), and
• the convex functional test as the natural requirement in convex optimization, which reflects affine conjugacy (Section 3.4).
For each of these three affine invariance classes, adaptive trust region strategies
are designed in view of an efficient choice of damping factors in Newton’s
method. They are all based on the paradigm already mentioned at the end
of Section 1.2. On a theoretical basis, details of algorithmic realization in
combination with either direct or iterative linear solvers are worked out. As
it turns out, an efficient determination of the steplength factor in global
Newton methods is intimately linked with the accuracy matching for affine
invariant combinations of inner and outer iteration within various inexact
Newton methods.
3.1 Globalization Concepts
Efficient iterative methods should be able to cope with ‘bad’ guesses x0 . In
this section we survey methods that permit rather general initial guesses x0 ,
not only those sufficiently close to the solution point x∗ . Of course, such
methods should merge into local Newton techniques as soon as the iterates
xk come ‘close to’ the solution point x∗ —to exploit the local quadratic or
superlinear convergence property.
3.1 Globalization Concepts
111
Parameter continuation methods. The simplest way of globalization of
local Newton methods is to embed the given problem F (x) = 0 into a oneparameter family of problems, a so-called homotopy
F (x, τ ) = 0 ,
τ ∈ [0, 1] ,
such that the starting point x0 is the solution for τ = 0 and the desired
solution point x∗ is the solution point for τ = 1. If we choose sufficiently
many intermediate problems in the discrete homotopy
F (x, τν ) = 0 ,
0 = τ0 < · · · < τν < · · · < τN = 1 ,
then the solution point of one problem can serve as initial guess in a local
Newton method for the next problem. In this way, global convergence can
be assured under the assumption that existence and uniqueness of the solution along the homotopy path is guaranteed. In this context, questions like
the adaptive choice of the stepsizes Δτν or the computation of bifurcation
diagrams are of interest. An efficient choice of the embedding will exploit
specific features of the given problem to be solved—with consequences for
the local uniqueness of the solution along the homotopy path and for the
computational speed of the discrete continuation process. The discussion of
these and many related topics is postponed to Chapter 5.
Pseudo-transient continuation methods. Another continuation method
uses the embedding of the algebraic equation into an initial value problem of
the type
x = F (x) ,
x(0) = x0 .
Discretization of this problem with respect to a timestep τ by the explicit
Euler method leads to the fixed point iteration
xk+1 − xk = Δxk = τ F (xk )
or, by the linearly implicit Euler method to the iteration scheme
I − τ F (xk ) Δxk = τ F (xk ) .
Note that for τ → ∞ the latter scheme merges into the ordinary Newton
method. The scheme reflects affine similarity as described in Section 1.2 and
will be treated in detail in Section 6.4 in the context of so-called pseudotransient continuation methods, which are a special realization of stiff integrators for ordinary differential equations.
3.1.1 Componentwise convex mappings
The Newton-Raphson method for scalar equations (see Figure 1.1) may be
geometrically interpreted as taking the intersection of the local tangent with
112
3 Systems of Equations: Global Newton Methods
the axis—and repeating this process until sufficient accuracy is achieved. In
this interpretation, the simplified Newton method just means to keep the
initial tangent throughout the whole iterative process. From this it can be
directly seen that both the ordinary and the simplified Newton method converge globally for convex (or concave) scalar functions. The convergence is
monotone, i.e., the iterates xk approach the solution point x∗ from one side
only. On the basis of this insight we are now interested in a generalization
of such a monotonicity property to systems. General convex minimization
problems, which lead to gradient mappings F , will be treated in the subsequent Section 3.4. Here we concentrate on some componentwise convexity as
discussed in the textbook of J.M. Ortega and W.C. Rheinboldt [163].
Such componentwise convex mappings F may be characterized by one of the
following equivalent properties (let x, y ∈ D ⊆ Rn , D convex, λ ∈ [0, 1]):
F λx + (1 − λ)y ≤ λF (x) + (1 − λ)F (y) ,
(3.2)
F (y) − F (x)
F (y) − F (x) (y − x)
≥
≥
F (x)(y − x) ,
0.
(3.3)
Herein, the inequalities are understood componentwise. Since the objects of
interest will be the iterates, we miss affine covariance in the above formulation. In fact, Ortega and Rheinboldt show monotone convergence of the
ordinary Newton method under the additional assumption
F (z)−1 ≥ 0 , z ∈ D ,
(3.4)
which is essentially a global M -matrix property (cf. R.S. Varga [192]) for the
Jacobian. Upon combining the above three equivalent convexity characterizations with (3.4), we obtain the three equivalent affine covariant formulations
F (z)−1 F λx + (1 − λ)y ≤ F (z)−1 λF (x) + (1 − λ)F (y)
(3.5)
−1
F (x)
F (y) − F (x) ≥ y − x
(3.6)
−1
F (z)
F (y) − F (x) (y − x) ≥ 0 .
(3.7)
Note that these conditions cover any mapping F such that (3.2) up to (3.3)
together with (3.4) hold for AF and AF with some A ∈ GL(n).
Lemma 3.1 Let F : D −→ Rn be a continuously differentiable mapping
with D ⊆ Rn open and convex. Let this mapping satisfy one of the convexity
characterizations (3.5)-(3.7). Then the ordinary Newton iteration starting at
some x0 ∈ D converges monotonically and globally such that componentwise
x∗ ≤ xk+1 ≤ xk ,
k = 1, 2, . . . .
(3.8)
3.1 Globalization Concepts
113
Proof. For the ordinary Newton iteration, one obtains:
xk+1 − xk = −F (xk )−1 F (xk ) − F (xk−1 ) − F (xk−1 )(xk − xk−1 )
1
k −1
= −F (x )
F xk−1 + δ(xk − xk−1 ) − F (xk−1 ) (xk − xk−1 )dδ .
δ=0
Insertion of (3.7) for
z = xk , x = xk−1 , y = xk−1 + δ(xk − xk−1 ), xk − xk−1 = (y − x)/δ
leads to
xk+1 − xk ≤ 0 for k ≥ 1 .
In a similar way, one derives
xk+1 − x∗ = (xk+1 − xk ) + (xk − x∗ ) = F (xk )−1 F (x∗ ) − F (xk ) + xk − x∗ ,
which, by application of (3.6), supplies
xk+1 − x∗ ≥ 0 , k ≥ 0 .
The rest of the proof can be found in [163], p. 453.
Remark 3.1 An immediate generalization of this lemma is obtained by
allowing different inequalities for different components in (3.5) to (3.7)—
which directly leads to the corresponding inequalities in (3.8).
Note that the above results do not apply to the simplified Newton iteration,
unless n = 1: following the lines of the above proof, the application of (3.7)
here would lead to
xk+1 − xk ≤ −F (x0 )−1 F (xk−1 ) − F (x0 ) (xk − xk−1 ) .
In order to apply (3.7) once more, a relation of the kind
xk−1 − x0 = Θ · (xk − xk−1 )
for some Θ > 0 would be required—which will only hold in R1 .
In actual computation, the global monotone convergence property does not
require any control in terms of some monotonicity test. Only reasonable componentwise termination criteria need to be implemented. It is worth mentioning that this special type of convergence of the ordinary Newton method does
not mean global quadratic convergence: rather this type of convergence may
be arbitrarily slow, as can be verified in simple scalar problems—see Exercise
1.3. Not even an a-priori estimation for the number of iterations needed to
achieve a prescribed accuracy may be possible.
114
3 Systems of Equations: Global Newton Methods
Bibliographical Note. The above componentwise monotonicity results
are discussed in detail in the classical monograph [163] by J.M. Ortega and
W.C. Rheinboldt, there in not affine invariant form. In 1987, F.A. Potra and
W.C. Rheinboldt proved affine invariant conditions, under which the simplified Newton method and other Newton-like methods converge, see [172, 170].
3.1.2 Steepest descent methods
A desirable requirement for any iterative methods would be that the iterates
xk successively approach the solution point x∗ —which may be written as
xk+1 − x∗ < xk − x∗ , if xk = x∗ .
Local Newton techniques implicitly realize such a criterion under affine covariant theoretical assumptions, as has been shown in detail in the preceding
chapter. Global methods, however, require a substitute approach criterion,
which may be based on the residual level function
T (x) :=
1
2
F (x) 2 ≡ 1 F (x)T F (x) .
2
2
(3.9)
Such a function has the property
T (x)
= 0 ⇐⇒ x = x∗ ,
T (x)
> 0 ⇐⇒ x = x∗ .
(3.10)
In terms of this level function, the approach criterion may be formulated as
a monotonicity criterion
T (xk+1 ) < T (xk ) , if T (xk ) = 0 .
Associated with the level function are the so-called level sets
G(z) := {x ∈ D ⊆ Rn | T (x) ≤ T (z)} .
◦
Let G denote the interior of G. Then property (3.10) implies
x∗ ∈ G(x) , x ∈ D
and the monotonicity criterion may be written in geometric terms as
◦
◦
xk+1 ∈ G(xk ) , if G(xk ) = ∅ .
An intuitive approach based on this geometrical insight, which dates back
even to A. Cauchy [44] in 1847, is to choose the steepest descent direction as
the direction of the iterative correction—see Figure 3.1.
3.1 Globalization Concepts
115
xk
x∗
xk+1
Fig. 3.1. Geometric interpretation: level set and steepest descent direction.
This idea leads to the following iterative method:
:= − grad T (xk ) = −F (xk )T F (xk )
:= xk + sk Δxk
Δxk
xk+1
sk
>
(3.11)
(3.12)
0 : steplength parameter.
Figure 3.1 also nicely shows the so-called downhill property.
Lemma 3.2 Let F : D −→ Rn be a continuously differentiable mapping with
D ⊆ Rn . Dropping the iterative index k in the notation of (3.11), let Δx = 0.
Then there exists a μ > 0 such that
T (x + sΔx) < T (x) , 0 < s < μ .
(3.13)
Proof. Define ϕ(s) := T (x + sΔx). As F ∈ C 1 (D), one has ϕ ∈ C 1 (D1 ),
D1 ⊆ R1 . Then
ϕ (0) = grad T (x + s · Δx)T Δx s=0 = −Δx22 < 0 .
With ϕ ∈ C 1 , the result (3.13) is established.
Steplength strategy. This result is the theoretical basis for a strategy to
select the steplength in method (3.12). It necessarily consists of two parts: a
reduction strategy and a prediction strategy. The reduction strategy applies
whenever
T (xk + s0k Δxk ) > T (xk )
for some given parameter s0k . In this case, the above monotonicity test is
repeated with some
si+1
k
:=
κ · sik ,
i = 0, 1, . . . ,
κ<1
(typically κ = 1/2) .
116
3 Systems of Equations: Global Newton Methods
Lemma 3.2 assures that a finite number i∗ of reductions will ultimately lead to
a feasible steplength factor s∗k > 0. The prediction strategy applies, when s0k+1
must be selected—usually based on an ad-hoc rule that takes the steplength
history into account such as
min (smax , s∗k /κ) , if s∗k−1 ≤ s∗k
s0k+1 :=
.
(3.14)
s∗k else
The possible increase from sk to sk+1 helps to avoid inefficiency coming
from ‘too small’ local corrections. One may also aim at implementing an
optimal choice of sk out of a sequence of sample values—a strategy, which is
often called optimal line search. However, since sk may range from 0 to ∞, a
reasonable set of values to be sampled may be hard to define, if a sufficiently
large class of problems is to be considered.
Convergence properties. An elementary convergence analysis shows that
the iterative scheme (3.12) with steplength strategies like (3.14) converges
linearly even for rather bad initial guesses x0 —however, possibly arbitrarily
slow. Moreover, so-called ‘pseudo-convergence’ characterized by
F (x)T F (x)
‘small’
may occur far from the solution point due to local ill-conditioning of the
Jacobian matrix.
General level functions. In a large class of problems, the described difficulties are a consequence of the fact that the whole scheme is not affine
covariant so that the choice of T (x) as a level function appears to be rather
arbitrary. In principle, any level function
2
T (x|A) := 12 AF (x)2
(3.15)
with arbitrary nonsingular (n, n)-matrix A could be used in the place of T (x)
above. To make things worse, even though the direction of steepest descent
Δx is ‘downhill’ with respect to T (x), there nearly always exists a matrix
A such that Δx is ‘uphill’ with respect to T (x|A), as will be shown in the
following lemma.
Lemma 3.3 Let Δx = − grad T (x) denote the direction of steepest descent
with respect to the level function T (x) as defined in (3.9). Then, unless
F (x)Δx = χ · F (x) , for some χ < 0 ,
there exists a class of nonsingular matrices A such that
T (x + sΔx|A) > T (x|A) , 0 < s < ν ,
for some ν > 0.
(3.16)
3.1 Globalization Concepts
117
Proof. Let F = F (x), J = F (x), J = JJ T , A = AT A. Then
ΔxT grad T (x|A) = −F T J AF .
Now, choose
A := J + μyy T
with some μ > 0 to be specified later and y ∈ Rn such that
F T (J + I)y = 0 , but F T y = 0 .
Here the assumption (3.16) enters for any χ ∈ R1 . By definition, however,
the choice χ ≥ 0 is impossible, since (3.16) implies that
χ=−
J T F 2
< 0.
F 2
Hence, for the above choice of A, we obtain
ΔxT grad T (x|A) = −JF 22 + μ(F T y)2 .
Then the specification
μ > AF 22 /(F T y)2
leads to
ΔxT grad T (x|A) > 0 ,
which, in turn, implies the statement of the lemma.
Summarizing, even though the underlying geometrical idea of steepest descent methods is intriguing, the technical details of implementation cannot
be handled in a theoretically satisfactory manner, let alone in an affine covariant setting.
3.1.3 Trust region concepts
As already shown in Section 1.1, the ordinary Newton method can be algebraically derived by linearization of the nonlinear equation around the solution point x∗ . This kind of derivation supports the interpretation that the
Newton correction is useful only in a close neighborhood of x∗ . Far away from
x∗ , such a linearization might still be trusted in some ‘trust region’ around
the current iterate xk . In what follows we will present several models defining
such a region. For a general survey of trust region methods in optimization
see, e.g., the book [45] by A.R. Conn, N.I.M. Gould, and P.L. Toint.
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3 Systems of Equations: Global Newton Methods
Levenberg-Marquardt model. The above type of elementary consideration led K.A. Levenberg [143] and later D.W. Marquardt [147]) to suggest a
modification of Newton’s method for ‘bad’ initial guesses that merges into
the ordinary Newton method close to the solution point. Following the presentation by J.J. Moré in [152] we define a correction vector Δx (dropping
the iteration index k) by the constrained quadratic minimization problem:
F (x) + F (x)Δx2 = min
subject to the constraint
Δx2 ≤ δ
in terms of some prescribed parameter δ > 0, which may be understood to
quantify the trust region in this approach.
The trust region constraint may be treated by the introduction of a Lagrange
multiplier p ≥ 0 subject to
p Δx22 − δ 2 = 0,
which yields the equivalent unconstrained quadratic optimization problem
F (x0 ) + F (x0 )Δx22 + pΔx22 = min .
After a short calculation and re-introduction of the iteration index k, we then
end up with the Levenberg-Marquardt method:
k T k
F (x ) F (x ) + pI Δxk = −F (xk )T F (xk ) , xk+1 := xk + Δxk . (3.17)
The correction vector Δxk (p) has two interesting limiting cases:
p → 0+
: Δxk (0) = −F (xk )−1 F (xk ) , if F (xk ) nonsingular
p→∞
1
: Δxk (p) → − grad T (xk ) .
p
In other words: Close to the solution point, the method merges into the
ordinary Newton method; far from the solution point, it turns into a steepest
descent method with steplength parameter 1/p.
Trust region strategies for the Levenberg-Marquardt method. All
strategies to choose the parameter p or, equivalently, the parameter δ are
based on the following simple lemma.
Lemma 3.4 Under the usual assumptions of this section let Δx(p) = 0 denote the Levenberg-Marquardt correction defined in (3.17). Then there exists
a pmin ≥ 0 such that
T x + Δx(p) < T (x) , p > pmin .
3.1 Globalization Concepts
119
Proof. Substitute q := 1/p , 0 ≤ q ≤ ∞ , and define
ϕ(q) := T x + Δx(1/q) , ϕ(0) = T (x) .
Then
ϕ (0) = 0 , ϕ (0) < 0 .
Hence, there exists a qmax = 1/pmin such that
ϕ(q) < ϕ(0) , 0 < q < qmax .
The method looks rather robust, since for any p > 0 the matrix J T J + pI is
nonsingular, even when the Jacobian J itself is singular. Nevertheless, similar
as the steepest descent method, the above iteration may also terminate at
‘small’ gradients, since for singular J the right-hand side of (3.17) also degenerates. This latter property is often overlooked both in the literature and
by users of the method. Since the Levenberg-Marquardt method lacks affine
invariance, special scaling methods are often recommended.
Bibliographical Note. Empirical trust region strategies for the LevenbergMarquardt method have been worked out, e.g., by M.D. Hebden [118], by
J.J. Moré [152], or by J.E. Dennis, D.M. Gay, and R. Welsch [54]. The associated codes are rather popular and included in several mathematical software
libraries. However, as already stated above, these algorithms may terminate
at a wrong solution with small gradient. When more than one solution exists
locally, these algorithms might not indicate that. The latter feature is particularly undesirable in the application of the Levenberg-Marquardt method
to nonlinear least squares problems—for details see Chapter 4 below.
Affine covariant trust region model. A straightforward affine covariant reformulation of the Levenberg-Marquardt model would be the following
constrained quadratic optimization problem:
F (x0 )−1 F (x0 ) + F (x0 )Δx 2 = min
subject to the constraint
Δx2 ≤ δ0 .
(3.18)
This problem can easily be solved geometrically—as shown in Figure 3.2,
where the constraint (3.18) is represented by a sphere around the current
iterate x0 with radius δ0 . Whenever δ0 exceeds the length of the ordinary
Newton correction, which means that the constraint is not active, then Δx is
just the ordinary Newton correction—and the quadratic functional vanishes.
Whenever the constraint is active, then the direction of Δx still is the Newton
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3 Systems of Equations: Global Newton Methods
direction, but now with reduced steplength. This leads to the Newton method
with so-called damping
F (xk )Δxk = −F (xk ) , xk+1 := xk + λk Δxk ,
wherein the damping factor varies in the range 0 < λk ≤ 1.
x0 − J(x0 )−1 F (x0 )
x0
δ0
Fig. 3.2. Geometric interpretation: affine covariant trust region model.
Affine contravariant trust region model. An affine contravariant reformulation of the Levenberg-Marquardt model would lead to a constrained
quadratic optimization problem of the form:
F (x0 ) + F (x0 )Δx = min
subject to the constraint
F (x0 )Δx2 ≤ δ0 .
Once again, the problem can be solved geometrically by Figure 3.2: only the
terms of the domain space of F must be reinterpreted by the appropriate
terms in the image space of F . As a consequence, the Newton method with
damping is obtained again.
Damping strategies for Newton method. All strategies for choosing the
above damping factors λk are based on the following insight.
Lemma 3.5 Under the usual assumptions of this section let F (x) be nonsingular and F = 0. With Δx defined to be the Newton direction there exists
some μ > 0 such that
T (x + λΔx) < T (x) , 0 < λ < μ .
Proof. As before, let F = F (x), J = F (x) and define ϕ(λ) := T (x + λΔx),
which then yields
ϕ(0) = T (x) , ϕ (0) = (J T F )T Δx = −F T F = −2T (x) < 0 .
Among the most popular empirical damping strategies is the
3.1 Globalization Concepts
121
Armijo strategy [7]. Let Λk ⊂ 1, 12 , 14 , . . . , λmin denote a sequence such
that
T (xk + λk Δxk ) ≤ 1 − 12 λk T (xk ) , λ ∈ Λk
(3.19)
holds and define an optimal damping factor via
T (xk + λk Δxk ) = min T (xk + λΔxk ) .
λ∈Λk
In order to avoid overflow in critical examples, the above evaluation of T will
be sampled from the side of small values λ. In a neighborhood of x∗ , this
strategy will produce λ = 1. If λ < λmin would be required, the iteration
should be terminated with a warning. Unfortunately, the latter occurrence
appears quite frequently in realistic problems of scientific computing, especially when the arising Jacobian matrices are ill-conditioned. This failure is
a consequence of the fact that the choice T (x) for the level function destroys
the affine covariance of the local Newton methods—a consequence that will
be analyzed in detail in Section 3.3 below.
3.1.4 Newton path
All globalization techniques described up to now were based on the requirement of local monotonicity with respect to the standard level function
T (x) = T (x|I) as defined in (3.9). In this section we will follow a more general
approach, which covers general level functions T (x|A) for arbitrary nonsingular matrix A as defined in (3.15). The associated level sets are written
as
G(z|A) := x ∈ D ⊆ Rn | T (x|A) ≤ T (z|A) .
(3.20)
With this notation, iterative monotonicity with respect to T (x|A) can be
written in the form
◦
◦
xk+1 ∈ G(xk |A) , if G(xk |A) = ∅ .
We start from the observation that each choice of the matrix A could equally
well serve within an iterative method. With the aim of getting rid of this
somewhat arbitrary choice, we now focus on the intersection of all corresponding level sets:
G(x) :=
G(x|A) .
(3.21)
A∈GL(n)
By definition, the thus defined geometric object is affine covariant. Its nature
will be revealed by the following theorem.
Theorem 3.6 Let F ∈ C 1 (D), D ⊆ Rn , F (x) nonsingular for all x ∈ D.
∈ GL(n), let the path-connected component of G(x0 |A)
in x0 be
For some A
compact and contained in D. Then the path-connected component of G(x0 )
122
3 Systems of Equations: Global Newton Methods
as defined in (3.21) is a topological path x : [0, 2] → Rn , the so-called Newton
path, which satisfies
F x(λ) = (1 − λ)F (x0 ) ,
(3.22)
2
0
T x(λ)|A = (1 − λ) T (x |A) ,
(3.23)
dx
=
dλ
x(0) =
dx dλ λ=0
−F (x)−1 F (x0 ) ,
(3.24)
x0 , x(1) = x∗ ,
−F (x0 )−1 F (x0 ) ≡ Δx0 ,
=
(3.25)
where Δx0 is the ordinary Newton correction.
Proof. Let F0 = F (x0 ). In a first stage of the proof, level sets and their
intersection are defined in the image space of F using the notation
H(x0 |A) :=
y ∈ Rn | Ay22 ≤ AF0 22 ,
H(x0 ) :=
H(x0 |A) .
A∈GL(n)
Let σi denote the singular values of A and qi the eigenvectors of AT A such
that
n
AT A =
σi2 qi qiT .
i=1
Select those A with q1 := F0 /F0 2 , which defines the matrix set:
A := A ∈ GL(n)|AT A =
n
σi2 qi qiT , q1 = F0 /F0 2 .
i=1
Then every y ∈ R can be represented by
n
n
bj q j , bj ∈ R .
y=
j=1
Hence
n
Ay22 = y T AT Ay =
σi2 b2i ,
i=1
AF0 22 = σ12 F0 22 ,
which, for A ∈ A, yields
3.1 Globalization Concepts
y
H(x |A) =
0
123
!
n
σi2 b2i ≤ σ12 F0 22
,
i=1
or, equivalently, the n-dimensional ellipsoids
1
b2 +
F0 22 1
σ2
σ1 F0 2
2
b22
+ ··· +
σn
σ1 F0 2
2
b2n ≤ 1 .
For A ∈ A, all corresponding ellipsoids have a common b1 -axis of length
F0 2 —see Figure 3.3. The other axes are arbitrary.
b 2 , . . . , bn
b1
Fig. 3.3. Intersection of ellipsoids H(x0|A) for A ∈ A.
Figure 3.3 directly shows that
0 ) :=
H(x
H(x0 |A) = y = b1 q1 |b1 | ≤ F0 2
A∈A
=
=
y ∈ Rn |y = (1 − λ)F0 , λ ∈ [0, 2]
y ∈ Rn |Ay = (1 − λ)AF0 , λ ∈ [0, 2] , A ∈ GL(n) .
0 ). On the other hand, for y ∈ H(x
0 ), A ∈
As A ⊂ GL(n) : H(x0 ) ⊆ H(x
GL(n), one has
Ay22 = (1 − λ)2 AF0 22 ≤ AF0 22 ,
0 ) ⊆ H(x0 ) and, in turn, confirms
which implies H(x
0 ) = H(x0 ) .
H(x
The second stage of the proof now involves ‘lifting’ of the path H(x0 ) to
G(x0 ). This is done by means of the homotopy
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3 Systems of Equations: Global Newton Methods
Φ(x, λ) := F (x) − (1 − λ)F (x0 ) .
Note that
Φx = F (x) , Φλ = F (x0 ) .
), local continuation
Hence, Φx is nonsingular for x ∈ D. As D ⊃ G(x0 |A
0
starting at x(0) = x by means of the implicit function theorem finally establishes the existence of the path
) ⊂ D,
x ⊂ G(x0 |A
which is defined by (3.22) from Φ ≡ 0. The differentiability of x follows, since
F ∈ C 1 (D), which confirms (3.24) and (3.25).
The above theorem deserves some contemplation. The constructed Newton
path x̄ is outstanding in the respect that all level functions T (x|A) decrease
along x̄—this is the result (3.22). Therefore, a rather natural approach would
be to just follow that path computationally—say, by numerical integration of
the initial value problem (3.24). Arguments, why this is not a recommended
method of choice, will be presented in Section 5 in a more general context.
Rather, the local information about the tangent direction
Δx0
Δx0 should be used—which is just the Newton direction. In other words:
Even ‘far away’ from the solution point x∗ , the Newton direction is an outstanding direction, only its length may be ‘too large’ for highly nonlinear
problems.
Such an insight could not have been gained from the merely algebraic local
linearization approach that had led to the ordinary Newton method.
The assumptions in the above theorem deliberately excluded the case that
the Jacobian may be singular at some x
close to x0 . This case, however, may
and will occur in practice. Application of the implicit function theorem in
a more general situation shows that all Newton paths starting at points x0
will end at one of the following three classes of points—see the schematic
Figure 3.4:
• at the ‘nearest’ solution point x∗ , or
• at some sufficiently close critical point x
with singular Jacobian, or
• at some point on the boundary ∂D of the domain of F .
The situation is also illustrated in the rather simple, but intuitive Example
3.2, see Figure 3.10 below.
3.2 Residual Based Descent
125
∂D
Fig. 3.4. Newton paths starting at initial points x0 (•) will end at a solution
point x∗ (◦), at a critical point x
b (), or on the domain boundary ∂D.
Bibliographical Note. Standard derivations of the Newton path as a
mathematical object had started from the so–called continuous analog of
Newton’s method, which is the ODE initial value problem (3.23)—see, e.g.,
the 1953 paper [48] by D. Davidenko. The geometric derivation of the Newton
path from affine covariance as a ‘first principle’ dates back to the author’s
dissertation [59, 60] in 1972.
3.2 Residual Based Descent
In this section we study the damped Newton iteration
F (xk )Δxk = −F (xk ),
xk+1 = xk + λk Δxk ,
λk ∈]0, 1]
under the requirement of residual contraction
F (xk+1 ) < F (xk ) ,
which is certainly the most popular and the most widely used global convergence measure.
From Section 3.1.4 we perceive this iterative method as the tangent deviation from the Newton path, which connects the given initial guess x0 to the
unknown solution point x∗ —under sufficient regularity assumptions on the
Jacobian matrix, of course. The deviation is theoretically characterized by
means of affine contravariant Lipschitz conditions as defined in the convergence theory for residual based local Newton methods in Section 2.2.
In what follows, we derive theoretically optimal iterative damping factors and
prove global convergence within some range around these optimal factors
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3 Systems of Equations: Global Newton Methods
(Section 3.2.1). On this basis we then develop residual based trust region
strategies for the algorithmic choice of the damping factors. This is first done
for the exact Newton correction Δx as defined above (Section 3.2.2) and
second for an inexact variant using the iterative solver GMRES for the inner
iteration (Section 3.2.3).
3.2.1 Affine contravariant convergence analysis
From Lemma 3.5 above we already know that the Newton correction Δxk
points downhill with respect to the residual level function
T (x) := 12 F (x)22
and therefore into the interior of the associated residual level set
G(x) := {y ∈ D|T (y) ≤ T (x)} .
k
At a given iterate x , we are certainly interested to determine some steplength
(defined by the associated damping factor λk ) along the Newton direction
such that the residual reduction is in some sense optimal.
Theorem 3.7 Let F ∈ C 1 (D) with D ⊂ Rn open convex and F (x) nonsingular for all x ∈ D. Assume the special affine contravariant Lipschitz
condition
F (y) − F (x) (y − x) ≤ ωF (x)(y − x)2 for x, y ∈ D .
Then, with the convenient notation
hk := ωF (xk ) ,
and λ ∈ [0, min (1, 2/hk )] we have:
F (xk + λΔxk )2 ≤ tk (λ)F (xk )2 ,
where
tk (λ) := 1 − λ + 12 λ2 hk .
(3.26)
The optimal choice of damping factor in terms of this local estimate is
λk := min (1 , 1/hk ) .
Proof. Dropping the superscript index k we may derive
F (x + λΔx) = F (x + λΔx) − F (x) − F (x)Δx
=
λ
F (x + s · Δx) − F (x) Δx ds − (1 − λ)F (x)Δx s=0
≤ (1 − λ)F (x) + O(λ2 ) for λ ∈ [0, 1] .
(3.27)
3.2 Residual Based Descent
127
The arising O(λ2 )-term obviously characterizes the deviation from the Newton path and can be estimated as:
λ
F (x + s · Δx) − F (x) Δxds ≤ ω · 12 λ2 F (x)Δx2 = 12 λ2 hk · F (x) .
s=0
Minimization of the above defined parabola tk then directly yields λk with
the a-priori restriction to the unit interval due to the underlying Newton path
concept.
We are now ready to derive a global convergence theorem on the basis of this
local descent result.
Theorem 3.8 Notation and assumptions as in the preceding Theorem 3.7.
In addition, let D0 denote the path-connected component of G(x0 ) in x0 and
assume that D0 ⊆ D is compact. Let the Jacobian F (x) be nonsingular for
all x ∈ D0 . Then the damped Newton iteration (k = 0, 1, . . .) with damping
factors in the range
λk ∈ ε , 2λk − ε
and sufficiently small ε > 0, which depends on D0 , converges to some solution
point x∗ .
Proof. The proof is by induction using the local results of the preceding
theorem. In Figure 3.5, the estimation parabola tk defined in (3.26) is depicted
as a function of the damping factor λ together with the polygonal upper
bound
⎧
1
⎪
⎪
1 − 12 λ
, 0≤λ≤
,
⎨
hk
tk (λ) ≤
⎪
1
1
2
⎪
⎩ 1 + 12 λ −
,
≤λ≤
.
hk
hk
hk
Upon restricting λ to the range indicated in the present theorem, we immediately have
1
tk (λ) ≤ 1 − 12 ε , 0 < ε ≤
,
(3.28)
hk
which induces strict reduction of the residual level function T (x) in each
iteration step k. In view of a proof of global convergence, a question left to
discuss is whether there exists some global ε > 0. This follows from the fact
that
max F (x) < ∞
x∈D0
under the compactness assumption on D0 . Hence, whenever G(xk ) ⊆ D0 ,
then (3.28) assures that
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3 Systems of Equations: Global Newton Methods
1
tk
λ
ε
1
hk
2
hk
Fig. 3.5. Local reduction parabola tk together with polygonal upper bounds.
G xk+1 (λ) ⊂ G(xk ) ⊆ D0 .
With arguments similar as in the proof of Theorem 2.12, we finally conclude
by induction that the defined damped Newton iteration converges towards
some limit point x∗ with F (x∗ ) = 0, which completes the proof.
3.2.2 Adaptive trust region strategies
The above derived theoretical damping strategy (3.27) cannot be implemented directly, since the arising Kantorovich quantities hk are computationally unavailable due to the arising Lipschitz constant ω. The obtained
theoretical results can nevertheless be exploited for the construction of computational strategies. Following the paradigm of Section 1.2.3, we may determine damping factors in the course of the iteration as close to the convergence analysis as possible replacing the unavailable Lipschitz constants ω
by computational estimates [ω] and the unavailable Kantorovich quantities
hk = ωF (xk ) by computational estimates [hk ] = [ω]F (xk ). Such estimates can only be obtained by pointwise sampling of the domain dependent
Lipschitz constants, which immediately implies that
[ω] ≤ ω , [hk ] ≤ hk .
(3.29)
By definition, the estimates [·] will inherit the affine contravariant structure.
As soon as we have iterative estimates [hk ] at hand, associated estimates of
the optimal damping factors may be naturally defined:
λk := min (1, 1/[hk ]) .
(3.30)
The relation (3.29) induces the equivalent relation
[λk ] ≥ λk .
3.2 Residual Based Descent
129
This means that the estimated damping factors might be ‘too large’—
obviously an unavoidable gap between analysis and algorithm. As a consequence, repeated reductions might be necessary, which implies that any
damping strategy to be derived will have to split into a prediction strategy
and a correction strategy.
Bit counting lemma. As for the required accuracy of the computational
estimates, the following lemma is important.
Lemma 3.9 Notation as just introduced. Assume that the damped Newton
method with damping factors as defined in (3.30) is realized. As for the accuracy of the computational estimates let
0 ≤ hk − [hk ] < σ max 1, [hk ] for some σ < 1 .
(3.31)
Then the residual monotonicity test will yield
F (x k+1 ) ≤ 1 − 12 (1 − σ)λ F (xk ) .
Proof. We reformulate the relation (3.31) as
[hk ] ≤ hk < (1 + σ) max 1, [hk ] .
Then the above notation directly leads to the estimation
F (x k+1 )
F (xk )
≤
1 − λ + 12 λ2 hk
<
1 − λ + 12 (1 + σ)λ2 [hk ]
λ=[λk ]
λ=[λk ]
≤ 1 − 12 (1 − σ)λk .
For σ < 1, any computational estimates [hk ] are just required to catch the
leading binary digit of hk , in order to assure residual monotonicity. For σ ≤ 12 ,
we arrive at the restricted residual monotonicity test
F (xk+1 ) ≤ 1 − 14 λ F (xk ).
(3.32)
This test nicely compares with the Armijo strategy (3.19), though derived by
a different argument.
Computational estimates. After these preliminary considerations, we
now proceed to identify affine contravariant computational estimates [·]—
preferably those, which are cheap to evaluate in the course of the damped
Newton iteration. In order to derive such estimates, we first recall from Section 3.1.4 that the damped Newton method may be interpreted as a deviation
130
3 Systems of Equations: Global Newton Methods
from the associated Newton path. Measuring the deviation in an affine contravariant setting leads us to the bound
F (xk+1 ) − (1 − λ)F (xk ) ≤ 12 λ2 ωF (xk )2 ,
which, in turn, leads to the following lower bound for the affine contravariant
Kantorovich quantity:
[hk ] :=
2F (xk+1 ) − (1 − λ)F (xk )
≤ hk .
λ2 F (xk )
This estimate requires at least one trial value xk+1 = xk + λ0k Δxk so that
it can only be exploited for the design of a correction strategy of the kind
(i = 0, 1, . . .):
λi+1
:= min 12 λik , 1/[hi+1
(3.33)
k
k ] .
In order to construct a theoretically backed initial estimate λ0k , we may apply
the relation
F (xk+1 )
hk+1 =
hk ,
F (xk )
which directly inspires estimates of the kind
[h0k+1 ] =
F (xk+1 ) i∗
[h ] < [hik∗ ] ,
F (xk ) k
wherein i∗ indicates the final computable index within estimate (3.33) for the
previous iterative step k. Thus we are led to the following prediction strategy
for k ≥ 0:
λ0k+1 := min 1, 1/[h0k+1 ] .
As can be seen, the only empirical choice left to be made is the starting value
λ00 . It is recommended to set λ00 = 1 for ‘mildly nonlinear’ problems and
λ00 = λmin 1 for ‘highly nonlinear’ problems in a definition to be put in
the hands of the users.
Intermediate quasi-Newton steps. Whenever λk = 1 and the residual
monotonicity test yields
Θk =
F (xk+1 )
≤ Θmax < 1
F (xk )
for some default value Θmax , then the residual based quasi-Newton method
of Section 2.2.3 may be applied—compare Theorem 2.14. This means that
Jacobian evaluations are replaced by residual rank-1 updates. As for a possible
switch back from quasi-Newton steps to Newton steps just look into the
details of the informal quasi-Newton algorithm QNRES, also in Section 2.2.3.
The just described adaptive trust region strategy is realized in
3.2 Residual Based Descent
131
Algorithm NLEQ-RES. Set a required residual accuracy ε sufficiently above
the machine precision.
Guess an initial iterate x0 . Evaluate F (x0 ).
Set an initial damping factor either λ0 := 1 or λ0 1.
Norms are tacitly understood to be scaled smooth norms, such as D̄−1 · 2 ,
where D̄ is a diagonal scaling matrix, constant throughout the iteration.
For iteration index k = 0, 1, . . . do:
1. Step k:
Convergence test: If F (xk ) ≤ ε: stop. Solution found x∗ := xk .
Else: Evaluate Jacobian matrix F (xk ). Solve linear system
F (xk )Δxk = −F (xk )
For k > 0: compute a prediction value for the damping factor
λk := min(1, μk ),
μk :=
F (xk−1 ) μ
.
F (xk ) k−1
Regularity test: If λk < λmin : stop. Convergence failure.
2. Else: compute the trial iterate xk+1 := xk + λk Δxk and evaluate F (xk+1 ).
3. Compute the monitoring quantities
Θk :=
F (xk+1 )
,
F (xk )
μk :=
1
k
2 F (x )
F (xk+1 ) − (1 −
· λ2k
λk )F (xk )
If Θk ≥ 1 (or, if restricted: Θk > 1 − λk /4):
then replace λk by λk := min(μk , 12 λk ). Go to Regularity test.
Else:
let λk := min(1, μk ) .
If λk = λk = 1 and Θk < Θmax : switch to QNRES.
Else:
If λk ≥ 4λk : replace λk by λk and goto 2.
Else: accept xk+1 as new iterate. Goto 1 with k → k + 1.
3.2.3 Inexact Newton-RES method
In this section we discuss the inexact global Newton method
xk+1 = xk + λk δxk , 0 < λk ≤ 1
realized by means of GMRES such that (dropping the inner iteration index i)
F (xk )δxk = −F (xk ) + rk .
Let δxk0 = 0 and thus r0k = F (xk ). The notation here follows Section 2.2.4 on
local Newton-RES methods.
132
3 Systems of Equations: Global Newton Methods
Convergence analysis. Before going into details of the analysis, we want
to point out that the inexact Newton method with damping can be viewed
as a tangent step in xk for the inexact Newton path
F (
x(λ)) − rk = (1 − λ) F (xk ) − rk
or, equivalently,
F (
x(λ)) = (1 − λ)F (xk ) + λrk ,
(3.34)
k
∗
˙
wherein x
(0) = x , x
(0) = δx , but x
(1) = x . Hence, when approaching
x∗ , we will have to assure that rk → 0. With this geometric interpretation in
mind, we are now prepared to derive the following convergence statements.
k
Theorem 3.10 Under the assumptions of Theorem 3.7 for the exact Newton
iteration with damping, the inexact Newton-GMRES iteration can be shown to
satisfy
F (xk+1 )
Θk :=
≤ tk (λk , ηk )
(3.35)
F (xk )
with
tk (λ, η) = 1 − (1 − η)λ + 12 λ2 (1 − η2 )hk , ηk =
rk 2
< 1.
F (xk )2
The optimal choice of damping factor is
1
λk := min 1,
.
(1 + ηk )hk
(3.36)
Proof. Recall from Section 2.1.5 that for GMRES
F (xk ) − rk 22 = F (xk )22 − rk 22 = (1 − ηk2 )F (xk )22
for well-defined ηk < 1. Along the line xk + λδxk the descent behavior can
be estimated using
1
F (x + λδx ) = (1 − λ)F (x ) + λr + λ
k
k
k
k
F (xk + sλδxk ) − F (xk ) δxk ds .
s=0
The last right hand term is directly comparable to the exact case: in the
application of the affine contravariant Lipschitz condition we merely have to
replace Δxk by δxk and, accordingly,
F (xk )Δxk 2 = F (xk )2
by
F (xk )δxk 2 = (1 − ηk2 )F (xk )2 .
With this modification, the result (3.35) is readily verified. The optimal
damping factor follows from setting tk (λ) = 0. With λk ≤ 1 as restriction,
we have (3.36).
3.2 Residual Based Descent
133
Adaptive trust region method. In order to exploit the above convergence
analysis for the construction of an inexact Newton-GMRES algorithm, we will
follow the usual paradigm and certainly aim at defining certain damping
factors
1
λk := min 1,
(1 + ηk )[hk ]
in terms of affine contravariant computationally available estimates. First, if
we once again apply the ‘bit counting’ Lemma 3.9, we arrive at the inexact
variant of the restricted residual monotonicity test
1 − ηk
F (xk+1 )2 ≤ 1 −
λk F (xk )2 ,
4
which here replaces (3.32). Next, upon returning to the above proof of Theorem 3.10, we readily observe that
F (xk+1 ) − (1 − λk )F (xk ) − λk rk 2 ≤
λ2k
(1 − ηk2 )hk F (xk )2 .
2
On this basis, we may simply define the a-posteriori estimates
2F xk+1 (λ) − (1 − λ)F (xk ) − λrk 2
[hk ](λ) :=
≤ hk ,
λ2 (1 − ηk2 )F (xk )2
which give rise to the correction strategy (i = 0, 1, . . . , i∗k )
1
1 i
λi+1
=
min
λ
,
,
k
2 k
(1 + ηk )[hik ]
and the associated a-priori estimates
[h0k+1 ] := Θk [hik∗ ] ≤ hk+1 ,
which induce the prediction strategy (k = 0, 1, . . .)
1
λ0k+1 := min 1,
.
(1 + ηk+1 )[h0k+1 ]
As for the choice of ηk , we already have a strategy for λk = λk−1 = 1—see
Section 2.1.5. For λk < 1, some constant value ηk ≤ η with some sufficiently
small threshold value η can be selected (and handed over to the local Newton
method, see Section 2.2.4). Then only λ00 remains to be set externally.
The just described residual based adaptive trust region strategy in combination with the strategy to match inner and outer iteration is realized in the
code GIANT-GMRES.
134
3 Systems of Equations: Global Newton Methods
Bibliographical Note. Residual based inexact Newton methods date
back to R.S. Dembo, S.C. Eisenstat, and T. Steihaug [51]. Quite popular
algorithmic heuristics have been worked out by R.E. Bank and D.J. Rose
in [19] and are applied in a number of published algorithms. A different
global convergence analysis has been given in [90, 91] by S.C. Eisenstat and
H.F. Walker. Their strategies are implemented in the code NITSOL due to
M. Pernice and H.F. Walker [166]. They differ from the ones presented here.
3.3 Error Oriented Descent
In this section we study the damped Newton iteration
F (xk )Δxk = −F (xk ),
xk+1 = xk + λk Δxk ,
λk ∈]0, 1]
in an error oriented framework, which aims at overcoming certain difficulties that are known to arise in the residual based framework, especially in
situations where the Jacobian matrices are ill-conditioned—such as in discretized nonlinear partial differential equations. Once again, we treat the
damped Newton method as a deviation from the Newton path, but this time
we characterize the deviation by means of affine covariant Lipschitz conditions such as those used in the convergence theory for error oriented local
Newton methods in Section 2.1.
The construction of an error oriented globalization of local Newton methods
is slightly more complicated than in the residual based approach. For this
reason, we first recur to the concept of general level functions T (x|A) as already introduced in (3.15) for arbitrary nonsingular matrix A and study the
descent behavior of the damped Newton method for the whole class of such
functions in an affine covariant theoretical framework (Section 3.3.1). As it
turns out, the obtained theoretically optimal damping factors actually reflect
the observed difficulties of the residual based variants. Moreover, the analysis
directly leads to the specific choice A = F (xk )−1 , which defines the so-called
natural level function (Section 3.3.2). As a consequence for actual computation, the iterates are required to satisfy the so-called natural monotonicity
test
k+1
Δx < Δxk ,
wherein the simplified Newton correction Δx
F (xk )Δx
k+1
k+1
defined by
= −F (xk+1 )
is only computed to evaluate this test (and later also for an adaptive trust
region method). As for a proof of global convergence, only a theorem covering
a slightly different situation is available up to now—despite the convincing
global convergence properties of the thus derived algorithm! From the associated theoretically optimal damping factors we develop computational trust
3.3 Error Oriented Descent
135
region strategies—first for the exact damped Newton method as defined above
(Section 3.3.3) and second for inexact variants using error reducing linear iterative solvers for the inner iteration (Section 3.3.4).
3.3.1 General level functions
In order to derive an affine covariant or error oriented variant of the damped
Newton method we first recur to general level functions, which have been
already defined in (3.15) as
T (x|A) := 12 AF (x)22 .
Local descent. It is an easy task to verify that the Newton direction points
‘downhill’ with respect to all such level functions.
Lemma 3.11 Let F ∈ C 1 (D) and let Δx denote the ordinary Newton correction (dropping the iteration index k). Then, for all A ∈ GL(n),
ΔxT grad T (x|A) = −2T (x|A) < 0 .
This is certainly a distinguishing feature to any other descent directions—
compare Lemma 3.3. Hence, on the basis of first order information only,
all monotonicity criteria look equally well-suited for the damped Newton
method. The selection of a specific level function will therefore require second
order information.
Theorem 3.12 Let F ∈ C 1 (D) with D ⊂ Rn convex and F (x) = F (x)
nonsingular for all x ∈ D . For a given current iterate xk ∈ D let G(xk |A)
⊂ D for some A ∈ GL(n). For x, y ∈ D assume that
F (x)−1 F (y) − F (x) (y − x) ≤ ωy − x2 .
Then, with the convenient notation
hk := Δxk ω , hk := hk cond AF (xk )
one obtains for λ ∈ 0, min 1, 2/hk :
AF (xk + λΔxk ) ≤ tk (λ|A)AF (xk ) ,
(3.37)
tk (λ|A) := 1 − λ + 12 λ2 hk .
(3.38)
where
The optimal choice of damping factor in terms of this local estimate is
λk (A) := min 1 , 1/hk .
136
3 Systems of Equations: Global Newton Methods
Proof. Dropping the superscript index k we may derive
AF (x + λΔx) = A F (x + λΔx) − F (x) − F (x)Δx ⎛ λ
⎞
= A ⎝
F (x + δΔx) − F (x) Δxdδ − (1 − λ)F (x)Δx⎠
δ=0
≤ (1 − λ)AF (x) + O(λ2 ) for λ ∈ [0, 1] .
The arising O(λ2 )-term obviously characterizes the deviation from the Newton path and can be estimated as:
AF (x)
λ
F (x)−1 F (x + δΔx) − F (x) Δxdδ δ=0
−1
≤ AF (x)ω 12 λ2 Δx2 = 12 λ2 AF (x)hk AF (x)
AF (x)
−1
≤ 12 λ2 hk AF (x) AF (x)
AF (x) = 12 λ2 hk AF (x) .
Minimization of the parabola tk then directly yields λk (A) with the a-priori
restriction to the unit interval due to the underlying Newton path concept.
Global convergence. The above local descent result may now serve as a
basis for the following global convergence theorem.
Theorem 3.13 Notation and assumptions as in the preceding Theorem 3.12.
In addition, let D0 denote the path-connected component of G(x0 |A) in x0
and assume that D0 ⊆ D is compact. Let the Jacobian F (x) be nonsingular
for all x ∈ D0 . Then the damped Newton iteration (k = 0, 1, . . .) with damping
factors in the range
λk ∈ ε , 2λk (A) − ε
and sufficiently small ε > 0, which depends on D0 , converges to some solution
point x∗ .
Proof. The proof is by induction using the local results of the preceding
theorem. Moreover, it is just a slight modification of the proof of Theorem
3.8 for the residual level function. In particular, Figure 3.5 shows the same
type of estimation parabola tk as defined here in (3.38): once again, the proper
polygonal upper bound supplies the global upper bound
tk (λ|A) ≤ 1 − 12 ε , 0 < ε ≤
1
,
hk
(3.39)
3.3 Error Oriented Descent
137
which induces strict reduction of the general level function T (x|A) in each
iteration step k. We are now just left to discuss whether there exists some
global ε > 0. This follows from the fact that
max F (x)−1 F (x) · cond2 AF (x) < ∞
x∈D0
under the compactness assumption on D0 . Hence, whenever G(xk |A) ⊆ D0 ,
then (3.39) assures that
G xk+1 (λ)|A ⊂ G(xk |A) ⊆ D0 .
We therefore conclude by induction that
lim xk = x∗ ,
k→∞
which completes the proof.
Algorithmic limitation of residual monotonicity. The above theorem
offers an intriguing explanation, why the damped Newton method endowed
with the traditional residual monotonicity criterion
T (xk+1 |I) ≤ T (xk |I)
may fail in practical computation despite its proven global convergence property (compare Theorem 3.8): in fact, whenever the Jacobian is ill-conditioned,
then the ‘optimal’ damping factors are bound to satisfy
−1
λk (I) = hk cond2 F (xk )
< λmin 1 .
(3.40)
Therefore, in worst cases, also any computational damping strategies (no
matter, how sophisticated they might be) will lead to a practical termination
of the iteration, since then xk+1 ≈ xk , which means that the iteration ‘stands
still’. For illustration of this effect see Example 3.1 below, especially Fig. 3.9.
Another side of the same medal is the quite often reported observation that
for ‘well-chosen’ initial guesses x0 residual monotonicity may be violated
over several initial iterative steps even though the ordinary Newton iteration
converges when allowed to do so by skipping the residual monotonicity test.
In fact, from the error oriented local convergence analysis of Section 2.1, one
would expect to obtain the optimal value λk = 1 roughly as soon as the
iterates are contained in the ‘neighborhood’ of the solution x∗ —say, as soon
as for some iterate hk < 1. A comparison with the above
theorem,
however,
shows that a condition of the kind hk = hk cond2 F (xk ) < 1 would be
required in the residual framework. The effect is illustrated by Example 3.1
at the end of Section 3.3.2.
138
3 Systems of Equations: Global Newton Methods
Summarizing, we have the following situation:
Combining any damping strategy with the residual monotonicity criterion
may have the consequence that mildly nonlinear problems actually ‘look like’
highly nonlinear problems, especially in the situation (3.40); as a consequence,
especially in the presence of ill-conditioned Jacobians, the Newton iteration
with damping tends to terminate without the desired result—despite an underlying global convergence theorem like Theorem 3.13 for A = I.
3.3.2 Natural level function
The preceding section seemed to indicate that ‘all level functions are equal’;
here we want to point out that ‘some animals are more equal than others’
(compare George Orwell, Animal Farm).
As has been shown, the condition number cond2 AF (xk ) plays a central
role in the preceding analysis, at least in the worst case situation. Therefore,
due to the well-known property
cond2 AF (xk ) ≥ 1 = cond2 I ,
the special choice
Ak := F (xk )−1
seems to be locally optimal as a specification
of the matrix
in the general level
function. The associated level function T x|F (xk )−1 is called natural level
function and the associated natural monotonicity test requires that
k+1
Δx
2 ≤ Δxk 2
(3.41)
in terms of the ordinary Newton correction Δxk and the simplified Newton
correction defined by
Δx
k+1
:= −F (xk )−1 F (xk+1 ) .
This specification gives rise to several outstanding properties.
Extremal properties. For A ∈ GL(n) the reduction factors tk (λ|A) and
the theoretical optimal damping factors λk (A) satisfy:
tk (λ|Ak )
λk (Ak )
= 1 − λ + 12 λ2 hk ≤ tk (λ|A)
= min (1 , 1/hk ) ≥ λk (A) .
An associated graphical representation is given in Figure 3.6.
3.3 Error Oriented Descent
139
tk
λ
Fig. 3.6. Extremal properties of natural level function: reduction factors
tk (λ|A) and optimal damping factors λk (A).
Steepest descent property. The steepest descent direction for T (x|A) in
xk is
T
− grad T (xk |A) = − AF (xk ) AF (xk ) .
With the specification A = Ak this leads to
Δxk = − grad T (xk |Ak ) ,
which means that the damped Newton method in xk is a method of steepest
descent for the natural level function T (x|Ak ).
Merging property. The locally optimal damping factors nicely reflect the
expected behavior in the contraction domain of the ordinary Newton method:
in fact, we have
hk ≤ 1 =⇒ λk (Ak ) = 1 .
Hence, quadratic convergence is asymptotically achieved by the damping
strategy based on λk .
Asymptotic distance function. For F ∈ C 2 (D), we easily verify that
T x|F (x∗ )−1 = 12 x − x∗ 22 + O(x − x∗ 3 ) .
Hence, for xk → x∗ , the natural monotonicity criterion asymptotically merges
into a desirable distance criterion of the form
˙ xk − x∗ 2 ,
xk+1 − x∗ 2 ≤
which is exact for linear problems. The situation is represented graphically in
Figure 3.7. Far away from the solution point, this nice geometrical property
survives in the form that the osculating ellipsoid to the level surface at the
current iterate turns out to be an osculating sphere.
140
3 Systems of Equations: Global Newton Methods
x∗
Fig. 3.7. Natural level sets: asymptotic distance spheres.
In the linear case, the Jacobian condition number represents the quotient
of the largest over the smallest half-axis of the level ellipsoid. In the nonlinear case, too, Jacobian ill-conditioning gives rise to cigar-shaped residual
level sets, which, in general, are distorted ellipsoids. Therefore, geometrically
speaking, the natural level function realizes some nonlinear preconditioning.
Local descent. Any damping strategy based on the natural monotonicity
test is sufficiently characterized by Theorem 3.12: just insert A = Ak into
(3.37) and (3.38), which then yields
Δx
k+1
≤ 1 − λ + 12 λ2 hk Δxk .
Global convergence. In the present situation, the above global convergence
theorem for general level functions, Theorem 3.13, does not apply, since the
choice Ak varies from step to step. In order to obtain an affine covariant
global convergence theorem, the locally optimal choice A = F (xk )−1 will
now be modelled by the fixed choice A = F (x∗ )−1 —in view of the asymptotic
distance function property.
Theorem 3.14 Let F : D −→ Rn be a continuously differentiable mapping
with D ⊆ Rn open convex. Assume that x0 , x∗ ∈ D with x∗ unique solution
of F in D and the Jacobian F (x∗ ) nonsingular. Furthermore, assume that
(I) F (x) is nonsingular for all x ∈ D,
(II) the path-connected component D0 of G x0 |F (x∗ )−1 in x0 is compact
and contained in D,
(III) the following affine covariant Lipschitz condition holds
∗ −1 F (x )
F (y) − F (x) (y − x) ≤ ω∗ y − x2 for y, x ∈ D ,
(IV) for any iterate xk ∈ D let h∗k := ω∗ Δxk · F (xk )−1 F (x∗ ) < ∞.
3.3 Error Oriented Descent
141
As the locally optimal damping strategy we obtain
λ∗k := min (1 , 1/h∗k ) .
Then any damped Newton iteration with iterative damping factors in the
range
λk ∈ ε , 2λ∗k − ε for 0 < ε < 1/h∗k
converges globally to x∗ .
Proof. In the proof of Theorem 3.12 we had for x = xk :
AF (x + λΔx) ≤ (1 − λ)AF (x) + O(λ2 ) .
For A = F (x∗ )−1 the O(λ2 )-term may now be treated differently as follows:
F (x∗ )−1
λ
F (x + δΔx) − F (x) Δxdδ ≤
δ=0
λ
ω∗ · δΔx2 dδ
δ=0
≤ ω∗ · 12 λ2 Δx · F (x)−1 F (x∗ ) · F (x∗ )−1 F (x)
= 12 λ2 h∗k F (x∗ )−1 F (x) .
On the basis of the thus modified local reduction property, global convergence
in terms of the above specified level function can be shown along the same
lines of argumentation as in Theorem 3.13. The above statements are just
the proper copies of the statements of that theorem.
Corollary 3.15 Under the assumptions of the preceding theorem with the
replacement of x∗ by an arbitrary z ∈ D0 in the Jacobian inverse and the
associated affine covariant Lipschitz condition
−1 F (z)
F (y) − F (x) (y − x) ≤ ω(z) y − x2 for x, y, z ∈ D0 ,
a local level function reduction of the form
2 T xk + λΔxk |F (z)−1 ≤ 1 − λ + 12 λ2 hk (z) T xk |F (z)−1
in terms of
hk (z) := Δxk ω(z) F (xk )−1 F (z)
and a locally optimal damping factor
λk (z) := min (1, 1/hk (z))
can be shown to hold. Assuming further that the used matrix norm is submultiplicative, then we obtain for best possible estimates ω(z) the extremal
properties
142
3 Systems of Equations: Global Newton Methods
F (xk )−1 F (z) ω(z) ≥ ω(xk )
and
hk (xk ) ≤ hk (z) , λk ≥ λk (z) , z ∈ D0 .
The corollary states that the locally optimal damping factors in terms of
the locally defined natural level function are outstanding among all possible globally optimal damping factors in terms of any globally defined affine
covariant level function. Our theoretical convergence analysis shows that we
may substitute the global affine covariant Lipschitz constant ω by its more
local counterpart ωk = ω(xk ) defined via
k −1 F (x )
F (x) − F (xk ) (x − xk ) ≤ ωk x − xk 2 for x, xk ∈ D0 . (3.42)
We have thus arrived at the following theoretically optimal damping strategy
for the exact Newton method
xk+1 = xk + λk Δxk ,
λk := min(1, 1/hk ),
hk = ωk Δxk .
(3.43)
We must state again that this Newton method with damping based on the
natural monotonicity test does not have the comfort of an accompanying
global convergence theorem. In fact, U.M. Ascher and M.R. Osborne in [10]
constructed a simple example, which exhibits a 2–cycle in the Newton method
when monitored by the natural level function. Details are left as Exercise 3.3.
However, as shown in [33] by H.G. Bock, E.A. Kostina, and J.P. Schlöder,
such 2–cycles can be generally avoided, if the theoretical optimal steplength
λk is restricted such that λhk ≤ η < 1. Details are left as Exercise 3.4.
This restriction does not avoid m–cycles for m > 2—which still makes the
derivation of a global convergence theorem solely based on natural monotonicity impossible. Numerical experience advises not to implement this kind
of restriction—generically it would just increase the number of Newton iterations required.
Geometrical interpretation. This strategy has a nice geometrical interpretation, which is useful for a deeper understanding of the computational
strategies to be developed in the sequel. Recalling the derivation in Section
3.3.1, the damped Newton method at some iterate xk continues along the
tangent of the Newton path G(xk ) with effective correction length
xk+1 − xk = λk Δxk ≤ ρk := 1/ωk .
Obviously, the radius ρk characterizes the local trust region of the linear
Newton model around xk . The situation is represented schematically in Figure 3.8.
3.3 Error Oriented Descent
143
xk+1
xk
ρk
G(xk+1 )
G(xk )
x∗
Fig. 3.8. Geometrical interpretation: Newton path G(xk ), trust region around
xk , and Newton step with locally optimal damping factor λk .
Interpretation via Jacobian information. In terms of a relative change
of the Jacobian matrix we may write
k −1 k+1
*
F (x )
F (x
) − F (xk ) Δxk Δxk ≤ λk hk ≤ 1 .
This relation suggests the interpretation that Jacobian information at the
center xk of the trust region ball is valid along the Newton direction up to the
surface of the ball, which is xk+1 . Of course, such an interpretation implicitly
assumes that the maximum change actually occurs at the most distant point
on the surface—this property certainly holds for the derived upper bounds.
Beyond the trust region the Jacobian information from the center xk is no
longer useful, which then implies the construction of a new Newton path
G(xk+1 ) and the subsequent continuation along the new tangent—see once
again Figure 3.8.
Behavior near critical points. Finally, we want to discuss the expected
behavior of the Newton method with damping in the presence of some closeby critical point, say x
with singular Jacobian F (
x). In this situation, the
Newton path and, accordingly, the Newton iteration with optimal damping
will be attracted by x
. Examples of such a behavior have been observed fairly
often—in particular, when multiple solutions are separated by manifolds with
singular Jacobian, compare, e.g., Figure 3.10. Nevertheless, even in such a situation, a structural advantage of the natural level function approach may play
a role: whereas points x
represent local minima of T (x|I), which will attract
iterative methods based on the residual monotonicity
test, they
show up as
local maxima of the natural level functions since T x
|F (
x)−1 is unbounded.
For this reason, the above theoretical damped Newton method tends to avoid
local minima of T (x) whenever they correspond to locally isolated critical
points.
144
3 Systems of Equations: Global Newton Methods
Deliberate rank reduction. In rare emergency cases only, a deliberate
reduction of the Jacobian rank (the so-called rank-strategy) turns out to be
helpful—which means the application of intermediate damped Gauss-Newton
steps. For details, see Section 4.3.5 below.
At the end of Section 3.3.1, we described the limitations of residual monotonicity in connection with Newton’s method for systems of equations with
ill–conditioned Jacobian. This effect can be neutralized by requiring natural
monotonicity instead, as can be seen from the following illustrative example.
Example 3.1 Optimal orbit plane change of a satellite around Mars.
This optimal control problem has been modeled by the space engineer
E.D. Dickmanns [86] at NASA. The obtained ODE boundary value problem has been treated by multiple shooting techniques (see [71, Sect. 8.6.2.]
and Section 7.1 below). This led to a system of n = 72 nonlinear equations
with a Jacobian known to be ill–conditioned. The results given here are taken
from the author’s dissertation [59, 60]. The problem is a typical representative
out of a large class of problems that ‘look highly nonlinear’, but are indeed
essentially ‘mildly nonlinear’ as discussed at the end of Section 3.3.1.
106
104
102
102
100
100
10−2
10−2
10−4
10−4
10−6
10−6
10−8
k
0
1
2
3
4
5
6
7
10−8
k
0
1
2
3
4
5
6
7
Fig. 3.9. Mars satellite orbit problem. Left: no convergence in residual norms
F (xk )22 (◦) or scaled residual norms Dk−1 F (xk )22 (•). Right: convergence in natural level function, ordinary Newton corrections Dk−1 Δxk 22 (•) versus simplified
k+1 2
Newton corrections Dk−1 Δx
2 (◦).
Figure 3.9 documents the comparative behavior of residual level functions
(with and without diagonal scaling) and natural level functions. The Newton
iteration has been controlled by scaled natural monotonicity tests
Dk−1 Δx
k+1 2
2
≤ Dk−1 Δxk 22 ,
k = 0, 1, . . . ,
3.3 Error Oriented Descent
145
as shown on the right; in passing, we note that the second Newton step has
been performed using ‘deliberate rank reduction’ as described just above. On
the left, the iterative values of the traditional residual level functions, both
unscaled and scaled, are seen to increase drastically for the accepted Newton
steps. Obviously, only the natural level functions reflect the ‘approach’ of the
iterates xk toward the solution x∗ .
Bibliographical Note. The algorithmic concept of natural level functions
has been suggested in 1972 by P. Deuflhard [59] for highly nonlinear problems
with ill-conditioned Jacobian. In the same year, linear preconditioning has
been suggested by O. Axelsson (see [11]) on a comparable geometrical basis,
but for the purpose of speeding up the convergence of iterative solvers.
3.3.3 Adaptive trust region strategies
The above derived theoretical damping strategy (3.43) cannot be implemented directly, since the arising Kantorovich quantities hk contain the computationally unavailable Lipschitz constants ωk , which are defined over some
domain D0 —in view of Figure 3.8, even a definition over some local trust
region would be enough. The obtained theoretical results can nevertheless
be exploited for the construction of computational strategies. Following our
paradigm in Section 1.2.3 again, we determine damping factors in the course
of the iteration as close to the convergence analysis as possible by introducing computationally available estimates [ωk ] and [hk ] = [ωk ]Δxk for the
unavailable theoretical quantities ωk and hk = ωk Δxk .
Such estimates can only be obtained by pointwise sampling of the domain
dependent Lipschitz constants, which immediately implies that
[ωk ] ≤ ωk ≤ ωk (z) ,
[hk ] ≤ hk ≤ hk (z)
(3.44)
compare Corollary 3.15. By definition, the estimates [·] will inherit the affine
covariant structure. Suppose now that we have certain estimates [hk ] at hand.
Then associated estimates of the optimal damping factors may be naturally
defined as
λk := min (1, 1/[hk ]) .
(3.45)
The above relation (3.44) gives rise to the equivalent relation
[λk ] ≥ λk .
Clearly, any computed estimated damping factors may be ‘too large’—which,
in turn, means that repeated reductions might be necessary. Therefore, any
damping strategy to be derived will have to split into a prediction strategy
and a correction strategy.
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3 Systems of Equations: Global Newton Methods
Bit counting lemma. The efficiency of such damping strategies will depend
on the required accuracy of the computational estimates—a question, which
is studied in the following lemma.
Lemma 3.16 Notation as just introduced. Assume that the damped Newton method with damping factors as defined in (3.45) is realized. As for the
accuracy of the computational estimates, let
O ≤ hk − [hk ] < σ max 1, [hk ] for some σ < 1.
(3.46)
Then the natural monotonicity test will yield
k+1
Δx
≤ 1 − 12 (1 − σ)λ Δxk .
Proof. We reformulate the relation (3.46) as
[hk ] ≤ hk < (1 + σ) max 1, [hk ] .
For [hk ] ≥ 1, the above notation directly leads to the estimation
k+1
Δx
k
Δx ≤
1 − λ + 12 λ2 hk
<
1 − λ + 12 (1 + σ)λ2 [hk ]
λ=[λk ]
λ=[λk ]
≤ 1 − 12 (1 − σ)λk .
The case [hk ] < 1 follows similarly.
The above lemma states that, for σ < 1, the computational estimates [hk ]
are just required to catch the leading binary digit of hk , in order to assure
natural monotonicity. For σ ≤ 12 , we arrive at the following restricted natural
monotonicity test
k+1
Δx
2 ≤ 1 − 14 λ Δxk 2 ,
(3.47)
which might be useful in actual computation to control the whole iterative
process more closely—compare also the residual based restricted monotonicity test (3.32) and the Armijo strategy (3.19).
Correction strategy. After these abstract considerations, we now proceed to derive specific affine covariant computational estimates [·]—preferably
those, which are cheap to evaluate in the course of the damped Newton iteration. For this purpose, we first recall the interpretation of the damped
Newton method as the tangent continuation along the Newton path as given
in Section 3.1.4. Upon measuring the deviation in an affine covariant setting,
we are led to the upper bound
Δx
k+1
(λ) − (1 − λ)Δxk ≤ 12 λ2 ωk Δxk 2 ,
3.3 Error Oriented Descent
147
which leads to estimates for the Kantorovich quantities
k+1
[hk ] = [ωk ]Δxk :=
2Δx
(λ) − (1 − λ)Δxk ≤ hk .
λ2 Δxk The evaluation of such an estimate requires at least one trial value λ0k (or
xk+1 , respectively). As a consequence, it can only be helpful in the design of
a correction strategy for the damping factor :
λj+1 := min 1 λ, 1/[hj ] (3.48)
j
k
2
k
λ=λk
for repetition index j = 0, 1, . . . .
Prediction strategy. We are therefore still left with the task of constructing
an efficient initial estimate λ0k . As it turns out, such an estimate can only be
gained by switching from the above defined Lipschitz constant ωk to some
slightly different definition:
F (xk )−1 F (x) − F (xk ) v ≤ ω k x − xk v ,
wherein the direction v is understood to be ‘not too far away from’ the
direction x − xk in order to mimic the above definition (3.42). With this
modified Lipschitz condition we may proceed to derive the following bounds:
k
Δx − Δxk = F (xk−1 )−1 − F (xk )−1 F (xk ) =
k
k
= F (xk )−1 F (xk ) − F (xk−1 ) Δx ≤ ωk λk−1 Δxk−1 · Δx .
This bound inspires the local estimate
k
[ω k ] :=
Δx − Δxk k
λk−1 Δxk−1 · Δx ≤ ωk ,
k
wherein, as required in the definition above, the direction Δx is ‘not too
far away from’ the direction Δxk−1 . In any case, the above computational
estimate exploits the ‘newest’ information that is available in the course of
the algorithm just before deciding about the initial damping factor. We have
thus constructed a prediction strategy for k > 0:
λ0k
:= min(1, μk ) ,
Δxk−1 k
Δx μk :=
·
· λk−1 .
k
k
Δx − Δxk Δx (3.49)
The only empirical choice left to be made is the starting value λ00 . In the public
domain code NLEQ1 (see [161]) this value is made an input parameter: if the
user classifies the problem as ‘mildly nonlinear’, then λ00 = 1 is set internally;
otherwise the problem is regarded as ‘highly nonlinear’ and λ00 = λmin 1
is set internally.
148
3 Systems of Equations: Global Newton Methods
Intermediate quasi-Newton steps. Whenever λk = 1 and the natural
monotonicity test yields
k+1
Θk =
Δx < 12 ,
Δxk then the error oriented quasi-Newton method of Section 2.1.4 may be applied in the present context—compare Theorem 2.9. In this case, Jacobian
evaluations are replaced by Broyden rank-1 updates. As for a possible switch
back from quasi-Newton steps to Newton steps just look into the details of
the informal quasi-Newton algorithm QNERR.
Termination criterion. Instead of the termination criterion (2.14) we may
here use its cheaper substitute
k+1
Δx
≤ XTOL .
k+1
Recall that then xk+2 = xk+1 + Δx
is cheaply available with an accuracy
of O(XTOL2 ).
The here described adaptive trust region strategy leads to the global Newton
algorithm NLEQ-ERR, which is a slight modification of the quite popular code
NLEQ1 [161].
Algorithm NLEQ-ERR. Set a required error accuracy ε sufficiently above the
machine precision.
Guess an initial iterate x0 . Evaluate F (x0 ).
Set a damping factor either λ0 := 1 or λ0 1.
All norms of corrections below are understood to be scaled smooth norms such
as D−1 · 2 , where D is a diagonal scaling matrix, which can be iteratively
adapted together with the Jacobian matrix.
For iteration index k = 0, 1, . . . do:
1. Step k: Evaluate Jacobian matrix F (xk ). Solve linear system
F (xk )Δxk = −F (xk ) .
Convergence test: If Δxk ≤ ε: stop. Solution found x∗ := xk + Δxk .
For k > 0: compute a prediction value for the damping factor
k
λk := min(1, μk ),
μk :=
Δxk−1 · Δx k
Δx − Δxk · Δxk · λk−1 .
Regularity test: If λk < λmin : stop. Convergence failure.
3.3 Error Oriented Descent
149
2. Else: compute the trial iterate xk+1 := xk + λk Δxk and evaluate F (xk+1 ).
Solve linear system (‘old’ Jacobian, ‘new’ right hand side):
F (xk )Δx
k+1
= −F (xk+1 ) .
3. Compute the monitoring quantities
k+1
Δx ,
Δxk Θk :=
μk :=
1
Δxk 2
k+1
Δx
· λ2k
− (1 − λk )Δxk .
If Θk ≥ 1 (or, if restricted: Θk > 1 − λk /4):
then replace λk by λk := min(μk , 12 λk ). Go to Regularity test.
Else:
let λk := min(1, μk ) .
If λk = λk = 1:
k+1
If Δx
≤ ε : stop.
k+1
Solution found x∗ := xk+1 + Δx
.
If Θk < 12 : switch to QNERR
Else:
If λk ≥ 4λk : replace λk by λk and goto 2.
Else:
accept xk+1 as new iterate.
Goto 1 with k → k + 1.
In what follows we want to demonstrate the main feature of this algorithm
at a rather simple, but very illustrative example for n = 2.
Example 3.2 [161]. The equations to be solved are
exp(x2+ y 2 ) − 3 = 0 ,
x + y − sin 3(x + y) = 0 .
For this simple problem, critical interfaces with singular Jacobian can be
calculated to be the straight line
y=x
and the family of parallels
y = −x ±
1
3
arccos
1
3
± 23 π · j , j = 0, 1, 2 . . . .
For illustration, the quadratic domain
−1.5 ≤ x , y ≤ 1.5
is picked out. This domain contains the six different solution points and five
critical interfaces.
150
3 Systems of Equations: Global Newton Methods
Computation of Newton paths. As derived in Section 3.1.4, the Newton
path x(λ), λ ∈ [0, 1] may be defined either by the homotopy
F x(λ) − (1 − λ)F (xk ) = 0
or by the initial value problem
F (x)
dx
= −F (xk ) , x(0) = xk .
dλ
This implicit ordinary differential equation can be solved numerically, say,
by implicit BDF codes [98] like DASSL [167] due to L. Petzold or by linearly
implicit extrapolation codes like LIMEX [75, 79]. In any such discretization,
linear subsystems of the kind
F (x)Δx − βΔλF (x)[F (xk )−1 F (xk ), Δx] = −ΔλF (xk )
must be solved. Apparently, this algorithmic approach involves second order derivative information in tensor form—to be compared with the above
described global Newton methods, which involve second order derivative information only in scalar form (Lipschitz constant estimates [ω] entering into
the adaptive trust region strategies). Note, however, that the Newton path
should be understood as an underlying geometric concept rather than an
object to be actually computed.
Fig. 3.10. Example 3.2: Newton paths (· · · ) versus Newton sequences (—).
Newton paths versus Newton sequences. Figure 3.10 shows the various
Newton paths (left upper part) and sequences of Newton iterates (right lower
part) as obtained by systematic variation of the initial guesses x0 —separated
by the symmetry line y = x. The Newton paths have been integrated by
3.3 Error Oriented Descent
151
LIMEX, whereas the Newton sequences have been computed by NLEQ1 [161].
As predicted by theory—compare Section 3.1.4 and Figure 3.4 therein—each
Newton path either ends at a solution point or at a critical point with singular
Jacobian. The figure clearly documents that this same structure is mimicked
by the sequence of Newton iterates as selected by the error oriented trust
region strategy.
Attraction basins. An adjacent question of interest is the connectivity
structure of the different attraction basins for the global Newton iteration
around the different solution points. In order to visualize these structures, a
rectangular grid of starting points (with grid size Δ = 0.06) has been defined
and the associated global Newton iteration performed.
Fig. 3.11. Example 3.2: attraction basins. Left: Global Newton method, code
NLEQ1 [161]. Right: hybrid method, code HYBRJ1 [153]. Outliers are indicated as
bullets (•).
The results are represented in Figure 3.11: apart from very few exceptional
‘corner points’, the attraction basins nicely model the theoretical connectivity structure, which is essentially defined by the critical interfaces—a highly
satisfactory performance of the herein advocated global Newton algorithm
(code NLEQ1 due to [161]). For comparison, the attraction basins for a hybrid
method (code HYBRJ1 in MINPACK due to [153]) are given as well. There are
still some people who prefer the rather chaotic convergence pattern of such
algorithms. However, in most scientific and engineering problems, a crossing
beyond critical interfaces is undesirable, because this means an unnoticed
switching between different solutions—an important aspect especially in the
parameter dependent case.
152
3 Systems of Equations: Global Newton Methods
3.3.4 Inexact Newton-ERR methods
Suppose that, instead of the exact Newton corrections Δxk , we are only
able to compute inexact Newton corrections δxk from (dropping the inner
iteration index i)
F (xk ) δxk − Δxk = rk , xk+1 = xk + λk δxk , 0 < λk ≤ 1 , k = 0, 1, . . . .
As for local inexact Newton-ERR methods (Section 2.1.5), we characterize
the inner iteration errors by the quantity
δk =
δxk − Δxk .
δxk (3.50)
Inner iterative solvers treated here are either CGNE or GBIT. As a guiding
principle for global convergence, we will focus on natural monotonicity (3.41)
subject to the perturbation coming from the truncation of the inner iteration.
Accuracy matching: inexact Newton corrections. First, we study the
contraction factors
k+1
Δx Θk (λ) =
Δxk in terms of the exact Newton corrections Δxk and the exact simplified Newton
k+1
corrections Δx
defined via
F (xk )Δx
k+1
= −F (xk + λδxk ) .
Note that the inexact Newton correction arises in the argument on the right.
Of course, none of the above exact Newton corrections will be actually computed.
Lemma 3.17 We consider the inexact Newton iteration with CGNE or GBIT
as inner iteration. Assume δk < 12 . Then, with hδk = ωδxk , we obtain the
estimate
δk
hδk
Θk (λ) ≤ 1 − 1 −
λ + 12 λ2
.
(3.51)
1 − δk
1 − δk
The optimal damping factor is
1 − 2δk
λk = min 1,
.
hδk
(3.52)
If we impose
ρ δ
λh , ρ ≤ 1 ,
2 k
we are led to the optimal damping factor
1
λk = min 1,
.
(1 + ρ)hδk
δk =
(3.53)
(3.54)
3.3 Error Oriented Descent
153
Proof. First, we derive the identity
λ
k+1
Δx
k −1
(λ) = Δx − λδx − F (x )
k
k
k
F (x + sδxk ) − F (xk ) δxk ds.
s=0
Upon inserting definition (3.50) and using the triangle inequality
δxk − Δxk δk
≤
,
k
Δx 1 − δk
δxk ≤
Δxk ,
1 − δk
the above estimate (3.51) follows directly. The optimal damping factors λk in
the two different forms arise by minimization of the upper bound parabola,
as usual.
Condition (3.53) is motivated by the idea that the O(λ) perturbation due to
the inner iteration should not dominate the O(λ2 ) term, which characterizes
the nonlinearity of the problem.
Accuracy matching strategy. Upon inserting λ = λk into (3.53) and selecting
some ρ ≤ 1, we are led to
δk ≤ δ̄ =
ρ
≤
2(1 + ρ)
1
4
for λk < 1
(3.55)
and
ρ δ
h for λk = 1 .
2 k
Of course, the realization of the latter rule will be done via computational
Kantorovich estimates [hδk ] ≤ hδk such that
δk ≤
δk ≤
ρ δ
[h ] for
2 k
λk = 1 .
(3.56)
Obviously, the relation (3.55) reflects the ‘fight for the first binary digit’ as
discussed in the preceding section; under this condition the optimal damping
factors (3.52) and (3.54) are identical. In passing we note that requirement
(3.56) nicely agrees with the ’quadratic convergence mode’ (2.62) in the local
Newton-ERR methods. (The slight difference reflects the different contraction
factors in the local and the global case.) The condition (3.56) is a simple
nonlinear scalar equation for an upper bound of δk .
As already mentioned at the beginning of this section, the exact natural
monotonicity test cannot be directly implemented within our present algorithmic setting. We will, however, use this test and the corresponding optimal
damping factor as a guideline.
154
3 Systems of Equations: Global Newton Methods
Accuracy matching: inexact simplified Newton corrections. In order
to construct an appropriate substitute for the nonrealizable Θk , we recur to
the inexact Newton path x
(λ), λ ∈ [0, 1], from (3.34), which satisfies
F (
x(λ)) = (1 − λ)F (xk ) + λrk .
Recall that the local inexact Newton correction δxk can be interpreted as
the tangent direction x
˙ (0) in xk . On this background, we are led to define a
perturbed (exact) simplified Newton correction via
+
F (xk )Δx
k+1
= −F (xk+1 ) + rk .
(3.57)
Lemma 3.18 With the notation and definitions of this section the following
estimate holds:
k+1
+
Δx
− (1 − λ)δxk ≤ 12 λ2 hδk δxk .
(3.58)
Proof. It is easy to verify the identity
λ
+ k+1 (λ) − (1 − λ)δxk = −F (xk )−1
Δx
F (xk + sδxk ) − F (xk ) δxk ds .
s=0
From this identity, the above estimate can be immediately derived in the
usual manner.
Of course, the linear equation (3.57) can only be solved iteratively. This
means the computation of an inexact simplified Newton correction satisfying
+i
F (xk )δx
k+1
= −F (xk+1 ) + rk + rik+1
for inner iteration index i = 0, 1, . . .. (In what follows, we will drop this index
wherever convenient.)
Initial values for inner iterations. In view of (3.58) we set the initial value
+ k+1 = (1 − λ)δxk .
δx
0
(3.59)
This means that the inner iteration has to recover only second order information. The same idea also supplies an initial guess for the inner iteration of
the inexact ordinary Newton corrections:
k
+ .
δxk0 = δx
(3.60)
This cross-over of initial values has proven to be really important in the
realization of any Newton-ERR method.
3.3 Error Oriented Descent
155
In what follows, we replace the nonrealizable contraction factor Θk (λ) by its
realizable inexact counterpart
k+1
+
k (λ) = δx Θ
δxk (3.61)
and study its dependence on the damping factor λ.
We start with CGNE as inner iterative solver.
Lemma 3.19 Notation as just introduced. Assume that the inner CGNE iteration with initial guess (3.59) has been continued up to some iteration index
i > 0 such that
k+1
k+1
+
+
Δx
− δx
i
ρi =
<1.
(3.62)
k+1
k+1
+
+
Δx
− δx
0
Then we obtain the estimate
k+1
+
δx
i
− (1 − λ)δxk ≤
1
2
1 − ρ2i λ2 hδk δxk .
(3.63)
Proof. In our context, the orthogonal decomposition (1.28) reads
k+1
+
Δx
k+1
+
− δx
0
k+1
+
2 = Δx
k+1
+
− δx
i
k+1
+
2 + δx
i
k+1
+
− δx
0
2 .
(3.64)
Insertion of (3.62) then leads to
+
(1 − ρ2i )Δx
k+1
k+1
+
− δx
0
k+1
+
2 = δx
i
k+1
+
− δx
0
2 .
(3.65)
With the insertion of (3.65) into (3.58) the proof is complete.
Observe that in CGNE the condition ρi < 1 arises by construction. The parameter ρi , however, is not directly computable from (3.62): the denominator
k+1
+
cannot be evaluated, since we do not have Δx
, but for the numerator a
rough estimate
k+1
k+1
+
+
i ≈ Δx
− δx
i
is available (see Section 1.4.3). Therefore we may define the computable parameter
k+1
k+1
+
+
Δx
− δx
i
i
ρi =
≈
.
(3.66)
k+1
k+1
k+1
k+1
+
+
+
+
δxi − δx0 δxi − δx
0
By means of (3.65), we then get
+
Δx
k+1
k+1
+
− δx
i
k+1
+
= ρi δx
i
k+1
+
− δx
0
= ρi
k+1
+
1 − ρ2i Δx
k+1
+
− δx
0
This result can be compared with (3.62) to supply the identification
.
156
3 Systems of Equations: Global Newton Methods
ρi = ρi /
1 − ρ2i
or ρi = ρi / 1 + ρ2i .
(3.67)
For GBIT as inner iterative solver, we also use ρi from (3.66), but in combination with a slightly different estimate.
Lemma 3.20 Notation as in the preceding lemma. Let ρi < 1 according to
(3.62). Then, for GBIT as inner iteration, we obtain
k+1
+
δx
i
− (1 − λ)δxk ≤
1 + ρi 2 δ
λ hk δxk .
2
(3.68)
Proof. We drop the iteration index i. For GBIT, we cannot do better than
apply the triangle inequality
k+1
k+1
+
δx
+
− (1 − λ)δxk ≤ Δx
k+1
+
− (1 − λ)δxk + δx
+
− Δx
k+1
.
With the requirement (3.62) we get
k+1
+
δx
k+1
+
− (1 − λ)δxk ≤ (1 + ρ)Δx
− (1 − λ)δxk .
Application of Lemma 3.18 then directly verifies the estimate (3.68).
(3.69)
Note that in GBIT the condition ρi < 1 is not automatically fulfilled, but
must be assured by the implementation. In order to actually estimate ρi , we
again recur to ρi from (3.66). If we combine (3.66) with (3.69), we now arrive
at
k+1
k+1
k+1
k+1
+
+
+
+
Δx
− δx
≤ ρi (1 + ρi )Δx
− δx
.
i
0
Comparison with (3.62) then supplies the identification
ρi = ρi /(1 + ρi ) or ρi = ρi /(1 − ρi ) for ρi < 1 .
(3.70)
Accuracy matching strategy. On the basis of the presented convergence analysis, we might suggest to run the inner iteration until
ρi ≤ ρmax
with
ρmax ≤
1
4
for both CGNE and GBIT. By means of the transformations (3.67) or (3.70),
respectively, this idea can be transferred to the realizable strategy
ρi ≤ ρmax .
(3.71)
Affine covariant Kantorovich estimates. Upon applying our algorithmic
paradigm from Section 1.2.3, we will replace the optimal damping factor λk
by computational estimates
3.3 Error Oriented Descent
1 − 2δk
1
[λk ] = min 1,
= min 1,
,
[hδk ]
(1 + ρ)[hδk ]
157
(3.72)
where [hδk ] = [ω]δxk ≤ hδk are Kantorovich estimates to be carefully selected.
For CGNE, we will exploit (3.63) thus obtaining the a-posteriori estimates
k+1
[hδk ]i
+
2δx
i
=
1−
k+1
+
− δx
0
2
2 2
k
ρi λ δx 2
≤ hδk .
Note that (3.64) assures the saturation property
[hδk ]i ≤ [hδk ]i+1 ≤ hδk .
Replacing ρi by ρi then gives rise to the computable a-posteriori estimate
[hδk ]i
≈
2
k+1
k+1
+
+
1 + ρ2i δx
− δx
i
0
λ2 δxk 2
2
.
(3.73)
For GBIT, we will exploit (3.68) and obtain the a-posteriori estimates
k+1
[hδk ]i =
k+1
+
+
2δx
− δx
i
0
≤ hδk .
2
(1 + ρi )λ δxk Here a saturation property does not hold. Replacement of ρi by ρi leads to
the computable a-posteriori estimates
[hδk ]i ≈
+ k+1 − δx
+k+1 2(1 − ρi )δx
i
0
.
λ2 δxk (3.74)
As for the construction of computational a-priori Kantorovich estimates, we
suggest to simply go back to the definitions and realize the estimate
[hδk+1 ] =
δxk+1 δ
[h ]∗ ,
δxk k
(3.75)
where [hδk ]∗ denotes the final estimate [hδk ]i from either (3.73) for CGNE or
(3.74) for GBIT, i.e. the estimate obtained at the final inner iteration step
i = ik of the previous outer iteration step k.
Bit counting lemma. Once computational estimates [hδk ] are available, we
may realize the damping strategy (3.72). In analogy to Lemma 3.16, we now
study the influence of the accuracy of the Kantorovich estimates.
158
3 Systems of Equations: Global Newton Methods
Lemma 3.21 Notation as just introduced. Let an inexact Newton-ERR
method with damping factors λ = [λk ] due to (3.72) be realized. Assume
that
1
0 ≤ hδk − [hδk ] < σ max
, [hδk ] for some σ < 1.
(3.76)
1+ρ
Then the exact natural contraction factor satisfies
k+1
Θk (λ) =
Δx 1
<1−
(1 − σ)λ .
Δxk 2+ρ
For CGNE, the inexact natural contraction factor is bounded by
+k+1 δx
11+σ
Θk =
<1− 1− 2
λ.
δxk 1+ρ
For GBIT, the inexact natural contraction factor is bounded by
k+1
+
k = δx < 1 − 1 − 1 (1 + ρ)(1 + σ) λ .
Θ
2
δxk 1+ρ
Proof. Throughout this proof, we will omit any results for λ = 1, since
these can be directly verified by mere insertion. This means that we assume
λ = [λk ] < 1 in the following.
For the exact natural monotonicity test we return to the inequality (3.51),
which reads
hδk
k ≤ 1 − 1 − 2δk λ + 1 λ2
Θ
.
2
2(1 − δk )
2(1 − δk )
Insertion of λ = [λk ] < 1 then yields
λhδk = (1 − 2δk )
hδk
1+σ
<
.
δ
1+ρ
[hk ]
Inserting this into the above upper bound and switching from the parameter
δk to ρ via (3.55) then verifies the first statement of the lemma.
For the inexact natural monotonicity test with CGNE as inner iterative solver,
we go back to (3.63), which yields
k (λ) ≤ 1 − λ + 1 1 − ρ2 λ2 hδ < 1 − λ + 1 λ2 hδ .
Θ
k
k
i
2
2
If we again insert the above upper bound λhδk , we arrive at
k < 1 − λ + 1 λ 1 + σ ,
Θ
2 1+ρ
which is equivalent to the second statement of the lemma.
3.3 Error Oriented Descent
159
For the inexact natural monotonicity test with GBIT as inner iterative solver,
we recur to (3.68), which yields
k (λ) ≤ 1 − λ + 1 + ρλ2 hδ .
Θ
k
2
Following the same lines as for CGNE now supplies the upper bound
k < 1 − λ + 1 + ρλ 1 + σ ,
Θ
2
1+ρ
which finally confirms the third statement of the lemma.
Inexact natural monotonicity tests. Suppose now that we require at
least one binary digit in the Kantorovich estimate, i.e., σ < 1 in Lemma 3.21.
In this case, exact natural monotonicity
k+1
Θk (λ) =
Δx
<1
Δxk would hold—which, however, is not realizable in the present algorithmic setting.
For CGNE, a computable substitute is the inexact natural monotonicity test
k+1
+
ρ
k = δx
Θ
<1−
λ.
k
δx 1+ρ
(3.77)
For GBIT, we similarly get
k+1
+
k = δx < 1 − ρ − ρλ .
Θ
δxk 1+ρ
(3.78)
k < 1;
The latter result seems to suggest the setting ρ ≤ ρ to assure Θ
otherwise inexact natural monotonicity need not hold.
Correction strategy. This part of the adaptive trust region strategy applies, if inexact natural monotonicity, (3.77) for CGNE or (3.78) for GBIT, is
violated. Then the damping strategy can be based on the a-posteriori Kantorovich estimates (3.73) for CGNE or (3.74) for GBIT, respectively, again written as [hδk ]∗ . In view of the exact correction strategy (3.48) with repetition
index j = 0, 1, . . ., we set
1
1
λj+1
:=
min
λ,
.
k
2
λ=λjk
(1 + ρ)[hδk ]∗
160
3 Systems of Equations: Global Newton Methods
Prediction strategy. This part of the trust region strategy is based on the
a-priori Kantorovich estimates (3.75). On the basis of information from outer
iteration step k − 1, we obtain
1
0
λk = min 1,
, k>0.
(1 + ρ)[hδk ]
For k = 0, we can only start with some prescribed initial value λ00 to be
chosen by the user.
If λ < λmin for some threshold value λmin 1 arises in the prediction or the
correction strategy, then the outer iteration must be terminated—indicating
some critical point with ill-conditioned Jacobian.
The described inexact Newton-ERR methods are realized in the programs
GIANT-CGNE and GIANT-GBIT with error oriented adaptive trust region strategy and corresponding matching between inner and outer iteration. The realization of the local Newton part here is slightly different from the one
suggested in the previous Section 2.1.5, since here we have the additional
+ available.
information of the inexact simplified Newton correction δx
Algorithms GIANT-CGNE and GIANT-GBIT.
1. Step k Evaluate F (xk ).
Solve linear system
F (xk )δxki = −F (xk ) + rik
for
i = 0, 1, . . . , ik
iteratively by CGNE or GBIT. Control of ik via accuracy matching strategy
(3.55) or (3.56).
If δxk ≤ XTOL: Solution: x∗ = xk + δxk .
Else: For k = 0: select λ0 ad hoc. For k > 0: determine λk = min 1,
strategy.
1
(1 + ρ)[hδk ]
from the prediction
Regularity test. If λk < λmin : stop
2. Else: compute trial iterate xk+1 := xk + λk δxk and evaluate F (xk+1 ).
Solve linear system
k+1
+
F (xk )δx
i
= −F (xk+1 ) + rk + rik+1
for i = 0, 1, . . . , ik
iteratively by CGNE or GBIT. Control of ik via accuracy matching strategy
(3.71).
3.4 Convex Functional Descent
161
Computation of Kantorovich estimates [hδk ].
3. Evaluate the monitor
k+1
+
k := δx 2 .
Θ
δxk 2
If monotonicity test (3.77) for CGNE or (3.78) for GBIT violated, then
refine λk according to correction strategy and go to regularity test.
Else go to 1.
As soon as the global Newton-ERR method approaches the solution point,
one may either directly switch to the local Newton-ERR methods presented
in Section 2.1.5 or merge the Kantorovich estimates from here with the ‘standard’, ‘linear’, or ‘quadratic’ convergence mode as described there.
Remark 3.2 The residual based algorithm GIANT-GMRES as presented in
Section 3.2.3 above seems to represent an easier implementable alternative to
the here elaborated error oriented algorithms GIANT-CGNE and GIANT-GBIT.
This is only true, if a ‘good’ left preconditioner CL is available. Indeed, if
spectral equivalence CL A ∼ I holds, then the preconditioned initial residual satisfies CL r0 ∼ x0 − x∗ . Otherwise, the here presented error oriented
algorithms realize some nonlinear preconditioning.
A numerical comparison of GIANT-CGNE, GIANT-GBIT, and NLEQ-ERR in the
context of discretized nonlinear PDEs is given in Section 8.2.1 below.
Bibliographical Note. The first affine covariant convergence proof for local inexact Newton methods has been given by T.J. Ypma [203]. A first error
oriented global inexact Newton algorithm has been suggested by P. Deuflhard
[67] on the basis of some slightly differing affine covariant convergence analysis. These suggestions led to the code GIANT by U. Nowak and coworkers
[160], wherein the inner iteration has been realized by an earlier version of
GBIT.
3.4 Convex Functional Descent
In the present section we want to minimize a general convex function f or,
equivalently, solve the nonlinear system F (x) = f T (x) = 0 with F (x) =
f (x) symmetric positive definite. It is not at all clear whether for general
functional the damped Newton method still is an efficient globalization. As
the damped Newton method can be interpreted as a tangent continuation
along the Newton path, we first study the behavior of an arbitrary convex
functional f along the Newton path x(λ) as a function of λ.
162
3 Systems of Equations: Global Newton Methods
Lemma 3.22 Let f ∈ C 2 (D) denote some strictly convex functional to be
minimized over some convex domain D ∈ Rn . Let F (x) = f (x) be symmetric positive definite in D and let x : [0, 1] → D denote the Newton path
starting at some iterate x(0) = xk and ending at the solution point x(1) = x∗
with F (x∗ ) = f T (x∗ ) = 0. Then f (x(λ)) is a strictly monotone decreasing
function of λ.
Proof. In the usual way we just verify that
λ
f (x(λ)) − f (x ) =
k
F (x(σ)), ẋ(σ) dσ .
σ=0
Insertion of (3.22) and (3.24) then leads to
λ
(1 − σ)F (x(σ))−1/2 F (xk )22 dσ
f (x(λ)) − f (x ) = −
k
σ=0
with a strictly positive definite integrand. Therefore, for 0 ≤ λ2 < λ1 ≤ 1:
λ1
f (x(λ1 )) − f (x(λ2 )) = −
(1 − σ)F (x(σ))−1/2 F (xk )22 dσ < 0 .
σ=λ2
Obviously, this result is the desired generalization of the monotone level function decrease (3.23). We are now ready to analyze the damped Newton iteration (k = 0, 1, . . .)
F (xk )Δxk = −F (xk ),
xk+1 = xk + λk Δxk ,
λk ∈]0, 1]
under the requirement of iterative functional decrease f (xk+1 ) < f (xk ).
3.4.1 Affine conjugate convergence analysis
As in the preceding sections, we first study the local reduction properties
of the damped Newton method within one iterative step from iterate xk to
iterate xk+1 .
Theorem 3.23 Let f : D → R1 be a strictly convex C 2 -functional to be
minimized over some open convex domain D ⊂ Rn . Let F (x) = f (x)T and
F (x) = f (x) symmetric and strictly positive definite. For x, y ∈ D, assume
the special affine conjugate Lipschitz condition
3.4 Convex Functional Descent
F (x)−1/2 (F (y) − F (x))(y − x) ≤ ωF (x)1/2 (y − x)2
163
(3.79)
with 0 ≤ ω < ∞. For some iterate xk ∈ D, define the quantities
k := F (xk )1/2 Δxk 22 , hk := ωF (xk )1/2 Δxk 2 .
Moreover, let xk + λΔxk ∈ D for 0 ≤ λ ≤ λkmax with
λkmax :=
4
1+
1 + 8hk /3
≤ 2.
Then
f (xk + λΔxk ) ≤ f (xk ) − tk (λ)k ,
(3.80)
tk (λ) = λ − 12 λ2 − 16 λ3 hk .
(3.81)
where
The optimal choice of damping factor is
λk =
1+
√
2
≤ 1.
1 + 2hk
(3.82)
Proof. Dropping the iteration index k, we apply the usual mean value theorem to obtain
f (x + λΔx) − f (x) =
−λ + 12 λ2 + λ2
1 1
s Δx, F (x + stλΔx) − F (x) Δx dtds .
s=0 t=0
Upon recalling the Lipschitz condition (3.79), the Cauchy-Schwarz inequality
yields
f (x + λΔx) − f (x) + λ − 12 λ2 ≤ λ3
1 1
s=0 t=0
s2 tF (x)1/2 Δx3 dtds = 16 λ3 h · ,
(3.83)
which confirms (3.80) and the cubic parabola (3.81). Maximization of tk by
tk = 0 and solving the arising quadratic equation then yields λk as in (3.84).
Moreover, by observing that
tk = λ 1 − 12 λ − 16 λ2 hk = 0
has only one positive root λkmax , the remaining statements are readily verified.
From these local results, we may easily proceed to obtain the following global
convergence theorem.
164
3 Systems of Equations: Global Newton Methods
Theorem 3.24 General assumptions as before. Let the path-connected component of the level set L0 := {x ∈ D | f (x) ≤ f (x0 )} be compact. Let
F (x) = f (x) be symmetric positive definite for all x ∈ L0 . Then the damped
Newton iteration (k = 0, 1, . . .) with damping factors in the range
λk ∈ [ε , min(1, λkmax − ε)]
and sufficiently small ε > 0, which depends on L0 , converges to the solution
point x∗ .
Proof. The proof just applies the local reduction results of the preceding
Theorem 3.23. The essential remaining task to show is that there is a common
minimal reduction factor for all possible arguments xk ∈ L0 . For this purpose,
just construct a polygonal upper bound for tk (λ) comparable to the polygon
in Figure 3.5. We then merely have to select ε such that
ε < min(λk , λkmax − λk )
for all possible iterates xk . Omitting the technical details, Figure 3.5 then
directly helps to verify that
f (xk + λΔxk ) ≤ (1 − γε)f (xk )
for λ in the above indicated range with some global γ > 0, which yields the
desired strict global reduction of the functional.
Summarizing, we have thus established the theoretical optimal damping strategy (3.82) in terms of the computationally unavailable Kantorovich quantities
hk .
Remark 3.3 It may be worth noting that the above analysis is nicely
connected with the local Newton methods (i.e., with λ = 1) as discussed in
Section 2.3.1. If we require that
λkmax =
1+
4
≥ 1,
1 + 8hk /3
then we arrive at the local contraction condition
hk ≤ 3.
This is exactly the condition that would have been obtained in the proof
of Theorem 2.18, if the requirement f (xk+1 ) ≤ f (xk ) had been made for
the ordinary Newton method—just compare (2.94). However, just as in the
framework of that section, the condition hk+1 ≤ hk also cannot be guaranteed
here, so that λk+1
max ≥ 1 is not assured. In order to assure such a condition,
the more stringent assumption hk < 2 as in (2.92) would be required.
3.4 Convex Functional Descent
165
3.4.2 Adaptive trust region strategies
Following our algorithmic paradigm from Section 1.2.3, we construct computational damping strategies on the basis of the above derived theoretically optimal damping strategy. This strategy contains the unavailable Kantorovich quantity hk , which we want to replace by some computational estimate [hk ] ≤ hk and, consequently, the theoretical damping factor λk defined
in (3.82) by some computationally available value
[λk ] :=
Since [hk ] ≤ hk , we have
2
1+
1 + 2[hk ]
≤ 1.
(3.84)
[λk ] ≥ λk
so that both a prediction strategy and a correction strategy need to be developed.
Bit counting lemma. As already observed in the comparable earlier cases,
the efficiency of such strategies depends on the required accuracy of the computational estimate, which we now analyze.
Lemma 3.25 Standard assumptions and notation of this section. Let
0 ≤ hk − [hk ] ≤ σ[hk ] for some σ < 1 .
(3.85)
Then, for λ = [λk ], the following functional decrease is guaranteed:
f (xk + λΔxk ) ≤ f (xk ) − 16 λ(λ + 2)k .
(3.86)
Proof. With hk ≤ (1 + σ)[hk ] and (3.83) we have (dropping the index k)
f (x + λΔx) − f (x) ≤ −tk (λ)k = −λ + 12 λ2 + 16 λ3 hk k
≤ −λ + 12 λ2 + 16 λ3 (1 + σ)[hk ] k .
At this point, recall that λk is a root of tk = 0 so that [λk ] is a root of
1 − λ − 12 λ2 [hk ] = 0 .
Insertion of the above quadratic term into the estimate then yields
f (x + λΔx) − f (x) ≤ −λ + 12 λ2 + 13 λ(1 + σ)(1 − λ) k .
Upon using σ < 1 (3.86) is confirmed.
(3.87)
The above functional monotonicity test (3.86) is suggested for use in actual
computation. If we further impose σ = 1/2 in (3.85), i.e., if we require at least
166
3 Systems of Equations: Global Newton Methods
one exact binary digit in the Kantorovich quantity estimate, then (3.87) leads
to the restricted functional monotonicity test
f (xk + λΔxk ) − f (xk ) ≤ − 12 λk .
We are now ready to discuss specific computational estimates [hk ] of the
Kantorovich quantities hk . Careful examination shows that we have three
basic cheap options. From (3.83) we have the third order bound
E3 (λ) := f (xk + λΔxk ) − f (xk ) + λ 1 − 12 λ k ≤ 16 λ3 hk k ,
which, in turn, naturally inspires the computational estimate
[hk ] :=
6|E3 (λ)|
≤ hk .
λ3 k
If E3 (λ) < 0, this means that the Newton method performs locally better
than for the mere quadratic model of f (equivalent to hk = 0). Therefore, we
decide to set
[λk ] = 1 , if E3 (λ) < 0 .
On the level of the first derivative we have the second order bound
E2 (λ) := Δxk , F (xk + λΔxk ) − (1 − λ)F (xk ) ≤ 12 λ2 hk k ,
which inspires the associated estimate
[hk ] :=
2|E2 (λ)|
≤ hk .
λ2 k
On the second derivative level we may derive the first order bound
E1 (λ) := Δxk , F (xk + λΔxk ) − F (xk ) Δxk ≤ λhk k ,
which leads to the associated estimate
[hk ] :=
|E1 (λ)|
≤ hk .
λk
Cancellation of leading digits in the terms Ei , i = 1, 2, 3 should be carefully
monitored—see Figure 3.12, where a snapshot at some iterate in a not further
specified illustrative example is taken. Even though the third order expression
is the most sensitive, it is also the most attractive one from the point of view
of simplicity. Hence, one should first try E3 and monitor rounding errors
carefully.
In principle, any of the above three estimates can be inserted into (3.84) for
[λk ] requiring at least one trial value of λ (or, respectively, xk+1 ). We have
therefore only designed a possible correction strategy
3.4 Convex Functional Descent
167
10000
1000
|E3 |
100
10
|E2 |
1
0.1
|E1 |
0.01
0.001
1e-09
1e-08
1e-07
1e-05
1e-06
0.01
0.001
0.0001
0.1
1
λ
Fig. 3.12. Computational Kantorovich estimates [hk ]: cancellation of leading
digits in |E3 |, |E2 |, |E1 |, respectively.
λi+1
:=
k
2
1+
1 + 2[hk (λ)]
λ=λik
.
(3.88)
In order to construct a theoretically backed initial estimate λ0k , we may recall
that hk+1 = Θk hk , where
1
Θk :=
F (xk+1 )− 2 F (xk+1 )2
1
F (xk )− 2 F (xk )2
.
This relation directly inspires the estimate
[h0k+1 ] := Θk [hi∗
k ,]
wherein i∗ indicates the final computable index within estimate (3.88) for the
previous iterative step k. Thus we are led to the following prediction strategy
for k ≥ 0:
2
λ0k+1 :=
≤ 1.
(3.89)
1 + 1 + 2[h0k+1 ]
As in the earlier discussed approaches, the starting value λ00 needs to be set
ad hoc—say, as λ00 = 1 for ‘mildly nonlinear’ problems and as λ00 = λmin 1
for ‘highly nonlinear’ problems.
168
3 Systems of Equations: Global Newton Methods
3.4.3 Inexact Newton-PCG method
On the basis of the above results for the exact Newton iteration, we may
directly proceed to obtain comparable results for the inexact Newton iteration
with damping (k = 0, 1, . . ., dropping the inner iteration index i)
F (xk ) δxk − Δxk = rk ,
xk+1 = xk + λk δxk ,
λk ∈ ]0, 1].
The inner PCG iteration is formally represented by the introduction of the inner residuals rk , which are known to satisfy the Galerkin condition (compare
Section 1.4).
δxk , rk = 0 .
(3.90)
The relative PCG error is denoted by
F (xk )1/2 (Δxk − δxk )
.
F (xk )1/2 δxk δk :=
We start the inner iteration with δxk0 = 0 so that (1.26) can be applied.
Convergence analysis. With this specification, we immediately verify the
following result.
Theorem 3.26 The statements of Theorem 3.23 hold for the inexact NewtonPCG method as well, if only the exact Newton corrections Δxk are replaced by
the inexact Newton corrections δxk and the quantities k , hk are replaced by
δk
:=
hδk
:=
k
,
1 + δk2
hk
ωF (xk )1/2 δxk =
.
1 + δk2
F (xk )1/2 δxk 2 =
Proof. Dropping the iteration index k, the first line of the proof of Theorem 3.23 may be rewritten as
f (x + λδx) − f (x) =
1
1 −λδ + 12 λ2 δ + λ2
s
δx, F (x + stλδx) − F (x) δx dtds + δx, r ,
s=0
t=0
wherein the last right hand term vanishes due to the Galerkin condition (3.90)
so that merely the replacement of Δx by δx needs to be performed.
With these local results established, we are now ready to formulate the associated global convergence theorem.
3.4 Convex Functional Descent
169
Theorem 3.27 General assumptions as Theorem 3.23 or Theorem 3.26,
respectively (in the latter case δk is formally assumed to be bounded). Let
the level set L0 := {x ∈ D | f (x) ≤ f (x0 )} be closed and bounded. Let
F (x) = f (x) be symmetric strongly positive for all x ∈ L0 . Then the damped
(inexact) Newton iteration (for k = 0, 1, . . .) with damping factors in the
range
λk ∈ [ε , min(1, λkmax − ε)]
and sufficiently small ε > 0 depending on L0 converges to the solution point
x∗ .
Proof. The proof just applies the local reduction results of the preceding
Theorem 3.23 or Theorem 3.26. The essential remaining task to show is that
there is a common minimal reduction factor for all possible arguments xk ∈
L0 . For this purpose, we simply construct a polygonal upper bound for tk (λ)
such that (omitting technical details)
f (xk + λΔxk ) ≤ f (xk ) − 12 εk
for λ in the above indicated range and all possible iterates xk with some
ε < min(λk , λkmax − λk ) .
This implies a strict reduction of the functional in each iterative step as
long as k > 0 and therefore global convergence in the compact level set L0
towards the minimum point x∗ with ∗ = 0.
Adaptive trust region strategy. The strategy as worked out in Section 3.4.2 can be directly copied, just replacing Δxk by δxk , k by δk , and hk
by hδk ; details are left to Exercise 3.5. In actual computation the orthogonality
condition (3.90) may be perturbed by rounding errors from scalar products
in PCG. Therefore the terms E3 and E2 should be evaluated in the special
form
E3 (λ) := f (xk + λδxk ) − f (xk ) − λ F (xk ), δxk − 12 λ2 δk
and
E2 (λ) := δxk , F (xk + λδxk ) − F (xk ) − λδk
with the local energy computed as δk = δxk , F (xk )δxk .
As for the choice of accuracies δk arising from the inner PCG iteration, we
once again require
δk ≤ 14
in the damping phase (λ < 1) and the appropriate settings as worked out
in Section 2.3.3 for the local Newton-PCG (λ = 1)—merging either into the
linear or the quadratic convergence mode.
The affine conjugate inexact Newton method with adaptive trust region
method and corresponding matching of the inner PCG iteration and outer
iteration is realized in the code GIANT-PCG.
170
3 Systems of Equations: Global Newton Methods
Bibliographical Note. The first affine conjugate global Newton method
has been derived and implemented by P. Deuflhard and M. Weiser [85], there
even in the more complicated context of an adaptive multilevel finite element
method for nonlinear elliptic PDEs—compare Section 8.3. The strategy presented here is just a finite dimensional analog of the strategy worked out
there.
Exercises
Exercise 3.1 Multipoint homotopy. Let F (x) = 0 denote a system of nonlinear equations to be solved and x∗ its solution. Let x0 , . . . , xp be a sequence of iterates produced by some iterative process. Consider the homotopy
(λ ∈ R1 )
p−1
Hp (x, λ) := F (x) −
Lk (λ)F (xk ) , p ≥ 1
k=0
with Lk being the fundamental Lagrangian polynomials defined over a set of
nodes 0 = λ0 < λ1 < · · · < λp = 1.
a) Show that, under the standard assumptions of the implicit function theorem, there exists a homotopy path x(λ) such that
x(λk ) = xk ,
k = 0, . . . , p − 1 ,
x(1) = x∗ .
Derive the associated Davidenko differential equation.
b) Construct an iterative process for successively increasing p = 1, 2, . . . by
appropriate discretization. What would be a reasonable assignment of the
nodes λ1 , . . . , λp−1 ? Consider the local convergence properties of such a
process.
c) Write a program for p = 1, 2 and experiment with λ1 over a set of test
problems.
Exercise 3.2 An obstacle on the way toward a proof of global convergence
for error oriented global Newton methods, controlled only by the natural
monotonicity test
F (xk )−1 F (xk+1 ) ≤ F (xk )−1 F (xk ) ,
is the fact that a desirable property like
F (xk+1 )−1 F (xk+1 ) ≤ F (xk )−1 F (xk )
does not hold.
(3.91)
Exercises
171
a) For xk+1 = xk + λΔxk , upon applying Theorem 3.12, verify that
F (xk+1 )−1 F (xk+1 ) ≤
1 − λ + 12 λ2 hk k −1
F (x ) F (xk ) .
1 − λhk
b) Show that only under the Kantorovich-type assumption hk < 1 the reduction (3.91) can be guaranteed for certain λ > 0.
Exercise 3.3 2–cycle example [10]. Consider a system F (x) = 0 of two
equations in two unknowns. Let
√
4√3 − 3
1
F (0) = − 10
=: −a , F (0) = I ,
−4 3 − 3
√
√ 4
17 3 −1/ 3
F (a) = 15
,
F (a) =
.
−3
1
1
Starting a Newton method with x0 := 0, we want to verify the occurrence of
a 2–cycle, if only natural monotonicity is required.
a) Show that in the first Newton step λ0 = 1 is acceptable, since the natural
monotonicity criterion is passed, which leads to x1 = a.
b) Show that in the second Newton step λ1 = 1 is acceptable yielding x2 =
x0 .
Exercise 3.4 Avoidance of 2–cycles [33]. We study the possible occurrence
of 2–cycles for a damped Newton iteration with natural monotonicity test and
damping factor λ. A special example is given in Exercise 3.3. By definition,
such a 2–cycle is characterized by the inequalities
F (xk )−1 F (xk+1 )
≤ F (xk )−1 F (xk ) ,
F (xk+1 )−1 F (xk+2 )
≤ F (xk+1 )−1 F (xk+1 )
with xk+2 = xk .
a) Upon applying Theorem 3.12 verify that
1 + λhk
k+1 −1
k+1
1 2
F (x
) F (x
) ≤ 1 − λ + 2 λ hk
F (xk+1 )−1 F (xk ) .
1 − λhk
b) Show that under the restriction
√
λhk ≤ η < 12 ( 17 − 3)
the occurrence of 2–cycles is impossible.
172
3 Systems of Equations: Global Newton Methods
c) By a proper adaptation of the bit counting Lemma 3.16, modify the
damping strategy (3.45) and the restricted monotonicity test (3.47) such
that 2–cycles are also algorithmically excluded.
Exercise 3.5 Consider an inexact Newton method for convex optimization,
where the inner iteration does not satisfy the Galerkin condition (3.90). The
aim here is to prove an affine conjugate global convergence theorem as a
substitute of Theorem 3.26. Define
σk = −
F (xk ), δxk .
δk
(a) Show that one obtains the upper bound
tk (λ) = σk λ − 12 λ2 − 16 λ3 hδk ,
so that σk > 0 is required to assert functional decrease.
(b) On the basis of the optimal damping factor
λk =
1+
2σk
1 + 2σk hδk
≤1
prove a global convergence theorem.
How can this theorem also be exploited for the design of an adaptive inexact
Newton method?
Exercise 3.6 Usual GMRES codes require the user to formulate the linear
equation Ay = b as AΔy = r(y0 ) with Δy = y − y 0 and r(y0 ) = b − Ay0 so
that Δy 0 = 0. Reformulate the linear system
+
F (xk )Δx
k+1
= −F (xk+1 ) + rk
to be solved by the GMRES iteration for an initial guess
k+1
+
δx
0
= (1 − λ)δxk .
4 Least Squares Problems: Gauss-Newton
Methods
In many branches of science and engineering so-called inverse problems arise:
given a series of system measurements and a conjectured model containing
unknown parameters, determine these parameters in such a way that model
and measurements ‘match best possible’. In this section, let x ∈ Rn denote
the parameters within a model function ϕ(t, x) to be determined from a comparison with given measurements (t1 , y1 ), . . . , (tm , ym ). For m > n, a perfect
match of model and data will not occur in general, caused by model deficiencies and/or measurement errors. Throughout this chapter, any vector norm
· will be understood to be the Euclidean norm.
Bibliographical Note. This kind of problem has first been faced, formulated, and solved by Carl Friedrich Gauss in 1795. His nowadays so-called
maximum likelihood method [96] has been published not earlier than 1809.
For an extensive appreciation of the historical scientific context see the recent
thorough treatise by A. Abdulle and G. Wanner [1]. The most elaborate recent survey about the numerical solution of least squares problems has been
published by Å. Bjørck [26].
Following the arguments of Gauss, the deviations between model and data
are required to satisfy a least squares condition of the type
2
m yi − ϕ(ti , x)
= min ,
(4.1)
δyi
i=1
wherein the δyi denote the error tolerances of the measurements yi . Proper
specification of the δyi within this problem is of crucial importance to permit
a reasonable statistical interpretation.
More generally, with
yi − ϕ(ti , x)
F = (fi (x)) =
, i = 1, 2, . . . ,
δyi
the above problem (4.1) appears as a special case of the following problem:
Given a mapping F : D ⊆ Rn → Rm with m > n, find a solution x∗ ∈ D
such that
F (x∗ )2 = min F (x) .
(4.2)
x∈D
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance
and Adaptive Algorithms, Springer Series in Computational Mathematics 35,
DOI 10.1007/978-3-642-23899-4_4, © Springer-Verlag Berlin Heidelberg 2011
173
174
4 Least Squares Problems: Gauss-Newton Methods
A nonlinear least squares problem is said to be compatible when
F (x∗ ) = 0 .
One way of attacking this problem is to reformulate it as a system of n
nonlinear equations
G(x) :=
1
2
grad F (x)2 = F (x)T F (x) = 0 .
Application of Newton’s method to G would require the solution of a sequence
of linear systems of the kind
F (x)T F (x) + F (x)T ◦ F (x) Δx = −F (x)T F (x) .
In a compatible problem, the second matrix term on the left will vanish at the
solution point. With the vague idea that the deviation between model and
data will not be ‘too large’, the tensor term F may be dropped to obtain
the so-called Gauss-Newton method:
F (x)T F (x)Δx = −F (x)T F (x) .
(4.3)
An alternative algorithmic approach directly starts from (4.2) using Taylor’s
expansion
F (x∗ ) = F (x0 ) + F (x0 )(x∗ − x0 ) + · · · .
Dropping quadratic terms as in the algebraic derivation of the ordinary Newton method, one ends up with the iterative method (k = 0, 1, . . .):
F (xk )Δxk + F (xk ) = min , xk+1 := xk + Δxk .
It is an easy task to verify that the solution of this local minimization problem
is again the Gauss-Newton method (4.3). Obviously, this method attacks the
solution of the nonlinear least squares problem by solving a sequence of linear
least squares problems.
As in Newton’s method for nonlinear equations (m = n), a natural distinction between local and global methods arises: Local Gauss-Newton methods
require ‘sufficiently good’ starting guesses x0 , whereas global Gauss-Newton
methods are constructed to handle ‘bad’ guesses as well; in contrast to the
nonlinear equation case (m = n), Gauss-Newton methods (for m > n) exhibit guaranteed convergence only for a subclass of nonlinear least squares
problems. In what follows, local and global Gauss-Newton methods for unconstrained and separable nonlinear least squares problems will be elaborated
in Sections 4.2 and 4.3, whereas the constrained case will only be treated in
Section 4.3—for theoretical reasons to be explained at the end of Section
4.1.2. Affine invariance of both theory and algorithms will once again play a
role, which here means affine contravariance and affine covariance.
Finally, in Section 4.4, we study underdetermined nonlinear systems, here
only in the affine covariant setting. As it turns out, a geodetic Gauss-Newton
path can be shown to exist generically and can be exploited to construct
a quasi-Gauss-Newton algorithm and a corresponding adaptive trust region
method.
4.1 Linear Least Squares Problems
175
Jacobian approximations. Before diving into details, a general warning
concerning Jacobian approximations within Gauss-Newton methods seems
to be in order. In contrast to Newton methods, Gauss-Newton methods may
be seriously affected by ‘too large’ Jacobian errors: the mathematical reason
for that is that they additionally carry information about local projections.
Therefore, analytic expressions for the Jacobian or automated differentiation
(see A. Griewank [112]) are strongly recommended.
4.1 Linear Least Squares Problems
In order to simplify the subsequent analysis of the nonlinear case, certain
notations and relations are first introduced here for the linear special case.
Bibliographical Note. For more details about linear least squares problems the reader may refer, e.g., to the textbooks by M.Z. Nashed [157] or by
A. Ben-Israel and T.N.E. Greville [24], wherein various generalized inverses
are characterized. As for the numerical linear algebra, the textbook [107] by
G.H. Golub and C.F. van Loan is still the classic. Moreover, the elaborate
quite recent handbook article [26] due to Å. Bjørck is a rich source.
4.1.1 Unconstrained problems
For given rectangular matrix A with m rows and n columns and given mvector y, we first consider the unconstrained linear least squares problem
Ax − y2 = min .
(4.4)
Recall that m > n and, in view of the intended data compression, typically
m n. The solution structure of this problem can be described as follows.
Lemma 4.1 Consider the linear least squares problem (4.4). Let
p := rank(A) ≤ n < m
denote the rank of the matrix A. Then the solution x∗ is unique if and only
if p = n. If p < n, then there exists an (n − p)-dimensional subspace X ∗ of
solutions. In particular, let x∗ denote the ‘shortest’ solution such that
x∗ ≤ x for all x ∈ X ∗ .
Then the general solution can be written as
x = x∗ + z
with arbitrary z ∈ N (A), the nullspace of A.
The proof can be found in any textbook on linear algebra (cf., e.g., [107]).
176
4 Least Squares Problems: Gauss-Newton Methods
Moore-Penrose pseudo-inverse. For 0 ≤ p ≤ n, the solution x∗ is unique
and can be formally written as
x∗ = A+ y
in terms of the Moore-Penrose pseudo-inverse A+ . This special generalized
inverse is uniquely defined by the so-called Penrose axioms:
(A+ A)T
=
A+ A ,
(AA+ )T
=
AA+ ,
+
+
A+ ,
A.
A AA
=
AA+ A =
From these axioms define the following orthogonal projectors emerge:
P := A+ A ,
P ⊥ := In − P ,
P := AA+ ,
P
⊥
:= Im − P .
The orthogonality properties of the projectors follow directly as:
P2 = P , PT = P
and
2
P =P, P
T
=P.
Let R(A) denote the range of the matrix A. Then P projects onto N ⊥ (A)
and P projects onto R(A). Since m > n, the relation
rank(A) = n ≤ m ⇐⇒ P = In , P ⊥ = 0
holds.
Inner and outer inverses. Beyond the Moore-Penrose pseudo-inverse A+ ,
a variety of other generalized inverses appear in realistic problems. In Section
4.2 below so-called inner inverses A− will play a role that satisfy only the
fourth of the above Penrose axioms:
A A− A = A .
(4.5)
Similarly, in Section 4.3, so-called outer inverses A− will play a role that
satisfy only the third of the above Penrose axioms:
A− A A− = A− .
(4.6)
Of course, the Moore-Penrose pseudo-inverse is both an inner and an outer
inverse.
4.1 Linear Least Squares Problems
177
Numerical solution. There are three basic approaches for the numerical
solution of linear least squares problems:
• QR-decomposition: This approach uses the fact that Ax − y =
Q(Ax − y) is invariant under orthogonal transformation Q; one will then
obtain
R
c
QA =
, Qy =
(4.7)
0
d
with R = (rij ) an upper triangular (n, n)-matrix. Formally speaking, whenever rank(A) = n, then R is nonsingular. In the above introduced notation
the residual norm of the solution is x∗ then simply
⊥
d = Ax∗ − y = P y .
In a similar way, we have the relation
c = Ax∗ = P y .
(4.8)
• Normal equations solution: This approach is the one originally derived
by Carl Friedrich Gauss who obtained
AT Ax = AT y
with symmetric positive semi-definite (n, n)-matrix AT A. Here (rational)
Cholesky decomposition will be the method of choice.
• Augmented system solution: In this approach one explicitly defines the
residual vector
r := y − Ax
as additional unknown, so that the normal equations can now be reformulated as
I
A
r
y
=
.
AT 0
x
0
This form is especially recommended for large sparse systems and for iterative
refinement—see [25].
Numerical rank decision. In actual computation, a precise rank determination for a given matrix A is hard, if not impossible. Instead a numerical
rank decision must be made, which depends on the selected linear solver as
well as on row and column scaling of the matrix—which includes the effect
of choosing the measurement tolerances δyi in (4.1). Since both the normal
equations and the augmented system approach essentially require full rank,
we elaborate here only on a slight modification of the QR-decomposition that
additionally applies column pivoting:
R
Q AΠ =
.
0
178
4 Least Squares Problems: Gauss-Newton Methods
Once again R = (rij ) is an upper triangular (n, n)-matrix. The matrix Π
represents column permutations performed such that
|r11 | ≥ |r22 | ≥ · · · ≥ |rnn | .
(4.9)
Let ε denote some reasonable input accuracy, then a numerical rank p may
be defined by the maximum index such that
ε|r11 | < |rpp | .
For p = n, the so-called subcondition number (cf. [83])
sc(A) :=
|r11 |
|rnn |
can be conveniently computed. Since
sc(A) ≤ cond2 (A) ,
rank-deficiency will certainly occur whenever
ε sc(A) ≥ 1 .
For p < n, a rank-deficient pseudo-inverse must be realized—e.g., by a QRCholesky decomposition due to [65] as presented in detail in Section 4.4.1
below.
Remark 4.1 An alternative, theoretically more satisfactory, but computationally more expensive numerical rank-decision is based on singular value
decomposition (cf. [106]).
4.1.2 Equality constrained problems
Consider the equality constrained linear least squares problem given in the
form
Bx − d2 = min
(4.10)
subject to
Ax − c = 0 ,
wherein A is an (m1 , n)-matrix, B an (m2 , n)-matrix, and c, d are vectors of
appropriate length with
m1 < n < m1 + m2 =: m .
There is a variety of well-known efficient methods to solve this type of problem numerically (see, e.g., the textbook [107]) so that there is no need to
repeat this material. However, in view of our later treatment of the constrained nonlinear least squares problems, we nevertheless present here some
derivation (brought to the knowledge of the author by C. Zenger) that will
play a fruitful role in the nonlinear case as well.
4.1 Linear Least Squares Problems
179
Penalty method. To start with we replace the constrained problem by a
weighted unconstrained problem of the kind
Tμ (x) := Bx − d2 + μ2 Ax − c2 = min ,
which can also be written in the form
μ(Ax − c)
Bx − d
2
= min .
The idea is to set the penalty parameter μ ‘sufficiently large’: however, the
associated formal solution
+ μA
μc
∗
xμ :=
B
d
should not be computed directly (see [190]), since
μA
pseudo-rank
−→ rank(A) for sufficiently large μ .
B
Penalty limit method. Fortunately, there exists a stable numerical variant
for
μ → ∞.
In order to see this, we apply simultaneous Householder transformations [43]
both to the above matrices and the right-hand sides—with μ finite, for the
time being. For ease of writing, only the first transformation is given: one has
to compute the quantities
,
m
1
σ(μ) := μ2
j=1
a2j,1 +
m
2
i=1
-1
2
b2i,1
,
−1
β(μ) := σ(μ) σ(μ) + μ|a1,1 |
,
,
l = 2, . . . , n :
2
yl (μ) := β(μ) μ
m
1
j=1
aj,1 aj,l +
m
2
bi,1 bi,l .
i=1
Then the effect of the Householder transformation on the matrices and vectors
is
μ · ajl → μ ajl + aj1 · yl (μ) , j = 1, . . . , m1 ,
i = 1, . . . , m2 ,
μ · cj
→ bil + bi1 · yl (μ) ,
→ μ cj + aj1 · y1 (μ) ,
di
→ di + bi,1 · y1 (μ) ,
i = 1, . . . , m2 .
bil
j = 1, . . . , m1 ,
These relations show that the limiting process μ → ∞ can be performed
yielding
180
4 Least Squares Problems: Gauss-Newton Methods
σ(μ)
σ
:= lim
=
μ→∞ μ
m
1
j=1
1
2
a2j,1
,
β := lim μ2 β(μ) = σ
(
σ + |a1,1 |)
−1
μ→∞
l = 2, . . . , n :
yl = lim yl (μ) = β
μ→∞
,
m
1
aj,1 aj,l
j=1
and eventually the transitions
aj,l
→ aj,l + aj,1 · yl , j = 1, . . . , m1 ,
bil
→ bil + bi1 · yl ,
i = 1, . . . , m2 ,
cj
→ cj + aj,1 · y1 ,
j = 1, . . . , m1 ,
di
→ di + aj,1 · y1 ,
i = 1, . . . , m2 .
Note that the transformation produces vanishing subdiagonal entries in the
matrices A and B. This type of transformation, however, is (column) orthogonal only with respect to A, but not with respect to B. Repeated application
of the transformations may be represented by the following scheme
n
m1
B
R1
Q1
m2
A
E
S1
R2
Q2
Here Q1 , Q2 are (column) orthogonal matrices and E describes an elimination
process producing zero entries in the associated part of the original matrix B.
R1 , R2 are upper triangular matrices with R1 being nonsingular, whenever
A is assumed to be of full rank m1 . Thus, if R2 is also nonsingular, then x∗∞
is bounded, since it is the unique solution of an upper triangular system with
triangular matrix
.
/
R1 S1
R :=
.
0
R2
Numerical rank decisions. If R2 is singular, then the associated rank
deficient Moore-Penrose pseudo-inverse must be taken. On the basis of this
type of decomposition, the two sub-condition numbers
sc(A) = sc(R1 ) and sc(R2 )
arise naturally. The first one, sc(R1 ), monitors the linear independence of
the equality constraints, whereas the second one, sc(R2 ), represents the constrained least squares problem as such. If both sub-condition numbers are
4.1 Linear Least Squares Problems
181
finite, then x∗∞ is certainly bounded. If sc(R1 ) > 1/eps for relative machine
precision eps, then m1 should be reduced. If sc(R2 ) > 1/eps, then a best
least squares solution can be aimed at. Again testing the condition number
via singular values would be safer, but more costly. Moreover, such an approach would spoil the nice structure as presented above. Details are clarified
by the subsequent lemma.
Lemma 4.2 In the above introduced notation, let
rank(A) = m1 .
Then,
x∗∞ := lim
μ→∞
μA
B
+ μc
d
is the (possibly shortest) solution of the above defined constrained linear least
squares problem.
Proof. The existence of some bounded x∗∞ has been shown above by deriving
an appropriate algorithm. It remains to be proven that x∗∞ is the solution
of the constrained problem (4.10). For this purpose, let μ < ∞ first. By
definition of x∗μ , one has
Tμ (x∗μ ) = Bx∗μ − d2 + μ2 Ax∗μ − c2
≤
Now, let
Bx − d2 + μ2 Ax − c2 = Tμ (x) .
(4.11)
S := x ∈ Rn |Ax − c = 0 .
Then, for any x ∈ S:
Tμ (x∗μ ) ≤ Bx − d2 .
(4.12)
On the other hand, (4.11) implies
Tμ (x∗μ ) ≥ μ2 Ax∗μ − c2 .
Combination of these two inequalities yields
Ax∗μ − c2 ≤
Bx − d2
,
μ2
which, for μ → ∞, shows that
x∗∞ ∈ S .
Hence
Tμ (x∗∞ ) = Bx∗∞ − d2 ,
which, together with (4.12) leads to
Bx∗∞ − d2 ≤ Bx − d2 for all x ∈ S .
This confirms the statement of the lemma.
182
4 Least Squares Problems: Gauss-Newton Methods
Projection limits. For later purposes, we want to point out that, by
columnwise application of Lemma 4.2, the domain space projector
+ μA
μA
P∞ := lim
B
B
μ→∞
exists, whereas the associated image space projector
+
μA
μA
P μ :=
B
B
would blow up in the limit μ → ∞. As a consequence, there exists a natural
nonlinear extension of the just presented penalty limit approach within the
error oriented framework (Section 4.3), but not within the residual oriented
one (Section 4.2).
4.2 Residual Based Algorithms
Upon recalling that the objective function (4.2) for unconstrained nonlinear
least squares problems is directly defined via the residual F (x), an affine contravariant approach to Gauss-Newton methods seems to be certainly natural.
This is the topic of the present section.
The class of iterations to be studied here is defined as follows: For given initial
guess x0 , let the sequence {xk } be defined by
Δxk := −F (xk )− F (xk ) , xk+1 = xk + λk Δxk ,
0 < λk ≤ 1 ,
(4.13)
where F : D ⊆ Rn → Rm and m > n. For λk = 1 we have the local and for
0 < λk ≤ 1 the global Gauss-Newton method. The generalized inverse F (x)−
is merely assumed to be an inner inverse (compare (4.5)) satisfying
F (x)F (x)− F (x) = F (x) ,
which, in terms of the projectors
P (x) := F (x)F (x)− ,
implies that
P (x)2 = P (x),
⊥
P (x) = Im − P (x)
P (x)F (x) = −F (x)Δx .
Note that P is the projector onto N ⊥ (F (x)), the orthogonal complement of
the nullspace of the Jacobian. If, in addition, the projector is symmetric
P (x)T = P (x) ,
then we may split the residual norm according to
4.2 Residual Based Algorithms
183
⊥
F (x)2 = P (x)F (x)2 + P (x)F (x)2 .
In any case, the minimum point x∗ of the objective function can be characterized by
P (x∗ )F (x∗ ) = 0 .
Note that for incompatible nonlinear least squares problems
⊥
F (x∗ ) = P (x∗ )F (x∗ ) = 0 .
4.2.1 Local Gauss-Newton methods
This section presents a direct extension of the affine contravariant convergence theory for local Newton methods as given in Section 2.2.1. Again affine
contravariant Lipschitz conditions of the type (1.8) will play an important
role. For the sake of clarity we once more point out that any such residual based analysis cannot be expected to touch upon the question of local
uniqueness of the solution.
Convergence analysis. In what follows we give a local convergence theorem
which is the direct extension of the affine contravariant Newton-Mysovskikh
theorem (Theorem 2.12) for Newton’s method. The generalized Jacobian inverse is only assumed to be an inner inverse, which makes the analysis applicable to a wider class of algorithms—exemplified below for unconstrained
and for separable Gauss-Newton algorithms.
Theorem 4.3 Let F : D ⊂ Rn → Rm with m > n be a differentiable mapping where D ⊂ Rn is open and convex. Let F (x)− denote an inner inverse of
the possibly rank-deficient Jacobian (m, n)-matrix and P (x) := F (x)F (x)−
a projector in the image space of F . At each iterate xk define the quantities hk := ωP (xk )F (xk ). Assume that the following affine contravariant
Lipschitz condition holds:
F (y) − F (x) (y − x) ≤ ωF (x)(y − x)2
for y − x ∈ N ⊥ (F (x)) and for x, y ∈ D .
Moreover, assume that
⊥
⊥
P (y)F (y) − P (x)F (x)| ≤ ρ(x)F (x)(y − x),
ρ(x) ≤ ρ < 1
for y − x ∈ N ⊥ (F (x)) and x, y ∈ D .
Define the open level set
2(1 − ρ) Lω = x ∈ D| P (x)F (x) <
ω
(4.14)
184
4 Least Squares Problems: Gauss-Newton Methods
and let L̄ω ⊂ D be compact. For a given initial guess x0 of an unknown
solution x∗ let
h0 = ωP (x0 )F (x0 ) < 2(1 − ρ) , i.e., x0 ∈ Lω .
(4.15)
Then the Gauss-Newton iterates {xk } defined by (4.13) remain in Lω and
converge to some solution point x∗ ∈ Lω with P (x∗ )F (x∗ ) = 0. The iterative
projected residuals {P (xk )F (xk )} converge to zero at an estimated rate
P (xk+1 )F (xk+1 ) ≤ (ρ + 12 hk )P (xk )F (xk ) .
(4.16)
Proof. For repeated induction, assume that xk ∈ Lω . The estimation to
follow may conveniently start from the identity
1
k+1
F (x
⊥
k
k
) = P (x )F (x ) +
k
F (x + sΔxk ) − F (xk ) Δxk ds .
s=0
From this, we may derive
⊥
P (xk+1 )F (xk+1 ) = F (xk+1 ) − P (xk+1 )F (xk+1 )
⊥
⊥
≤ P (xk+1 )F (xk+1 ) − P (xk )F (xk )
+
1
k
F (x + sΔxk ) − F (xk ) Δxk ds .
s=0
Since xk+1 − xk = Δxk ∈ N ⊥ F (xk ) , the first right hand term can be
estimated by means of assumption (4.14) to obtain
⊥
⊥
P (xk+1 )F (xk+1 ) − P (xk )F (xk ) ≤ ρP (xk )F (xk ) .
In a similar way the second term yields
1
k
F (x + sΔxk ) − F (xk ) Δxk ds ≤ 12 hk P (xk )F (xk ) .
s=0
Upon combining the two estimates, the result (4.16) is verified, which can be
rewritten as
hk+1
≤ ρ + 12 hk .
(4.17)
hk
Now, for the purpose of the induction proof, define
Lk := x ∈ D | P (x)F (x) ≤ P (xk )F (xk ) .
4.2 Residual Based Algorithms
185
Due to assumption (4.15) we have L0 ⊂ Lω . Now let Lk ⊂ Lω , then (4.17)
implies that hk+1 < hk so that Lk+1 ⊆ Lk ⊂ Lω , which completes the
induction. At this point we tacitly skip the usual contradiction argument to
verify that xk + λΔxk ∈ Lk for all λ ∈ [0, 1]—just compare the similar proof
of Theorem 2.12. As for the convergence to some (not necessarily unique)
solution point x∗ , arguments similar to those in the proof of Theorem 2.12
apply as well, mutatis mutandis, and are therefore omitted here.
Small residual problems. In the above convergence rate estimate (4.16)
the asymptotic convergence factor ρ is seen to play a crucial role: obviously,
ρ < 1 characterizes a class of nonlinear least squares problems that can be
solved by the Gauss-Newton method. For lack of a better name, we will adopt
the slightly misleading, but widespread name ‘small residual’ problems and
call the factor ρ also small residual factor.
Bibliographical Note. It has been known for quite a while that the
Gauss-Newton iteration fails to converge for so-called ‘large residual’ problems (compare, e.g., P.T. Boggs and J.E. Dennis [34] or V. Pereyra [165]).
As for their definition or even computational recognition, there is a variety
of options. In [101], P.E. Gill and W. Murray suggested to characterize such
problems by comparison of the singular values of the Jacobian, i.e., in terms
of first order derivative information. However, since linear problems converge
within one iteration—independent of the size of the residual or the singular
values of the Jacobian—the classification of a ‘large residual’ will necessarily
require second order information about the true nonlinearity of the problem.
The theory presented here asymptotically agrees with the beautiful geometric
theory of P.-Å. Wedin [196], which also uses second order characterization.
In fact, Theorem 4.3 implicitly contains the necessary condition
ρ(x∗ ) < 1 .
Following [196] this condition can be interpreted as a stability restriction on
the curvature of the manifold F (x) = F (x∗ ) —details are left as Exercise 4.2.
Unconstrained Gauss-Newton algorithm. We now proceed to exploit
the above convergence theory for the construction of a local Gauss-Newton
algorithm to solve unconstrained nonlinear least squares problems (4.2). In
this case we have the specification F (x)− = F (x)+ , i.e., the Moore-Penrose
inverse, so that the above theorem trivially applies.
Convergence monitor. Clearly, Theorem 4.3 suggests to introduce the contraction factors for the projected residuals
Θk :=
P (xk+1 )F (xk+1 )
F (xk+1 )Δxk+1 =
.
k
k
F (xk )Δxk P (x )F (x )
186
4 Least Squares Problems: Gauss-Newton Methods
Note that
Θk ≤ ρ + 12 hk < 1
characterizes the local convergence domain. Hence, divergence of the GaussNewton iteration is diagnosed, whenever Θk ≥ 1 arises. For compatible problems, quadratic convergence occurs. For incompatible problems, asymptotic
linear convergence will be observed.
Numerical realization. Numerical techniques for the realization of each iterative step of the unconstrained Gauss-Newton method have been given in
Section 4.1.1. Let Qk denote the orthogonal matrix such that
Rk
ck
Qk F (xk ) =
, Qk F (xk ) =
(4.18)
0
dk
with Rk an upper triangular (n, n)-matrix, assumed to be nonsingular here
for ease of presentation. Then, according to (4.8), the computation of Θk can
be realized in the simple form
Θk :=
ck+1 .
ck In order to distinguish the linear convergence phase, one may additionally
compute the modified contraction factor
Θ̂k :=
P (xk )F (xk+1 )
ĉk+1 =
ck P (xk )F (xk )
via the decomposition
k+1
Qk F (x
)=
ĉk+1
dˆk+1
.
(4.19)
It is an easy exercise to show that
Θ̂k ≤ 12 hk .
Termination criterion. On the basis of the above local convergence theory,
the iteration should be terminated at some point x̂ with
P (x̂)F (x̂)2 ≤ FTOL ,
where FTOL is a user prescribed residual error tolerance. An equivalent criterion is
|f (xk+1 ) − f (xk )| ≤ FTOL ,
where f (x) = F (x)22 denotes the least squares functional—compare also
the results of Exercise 4.1. The situation that the linear convergence behavior
dominates the quadratic one, may be recognized by Θ̂k Θk . There are good
statistical reasons to terminate the iteration then.
4.2 Residual Based Algorithms
187
Separable Gauss-Newton algorithms. In many applications of practical
interest part of the parameters to be fitted arise linearly, whereas others arise
nonlinearly. A typical representation would be
F (u, v) = min ,
where
F (u, v) := A(v)u − y ,
(4.20)
A(v) : (m, s)-matrix, u ∈ Rs , y ∈ Rm , v ∈ Rn−s .
Such problems are said to be separable. Typical examples occur in the analysis
of spectra, where the model function is just a series of Gauss or Lorentz
functions (interpreted as single spectral lines); then the variables u are the
amplitudes, whereas the variables v contain information about the phases
and widths of each of these bells.
Suppose now that the solution components v ∗ for the above problem were
already given. Then the remaining components u∗ would be
u∗ = A(v ∗ )+ y .
Hence
⊥
F (u∗ , v ∗ ) = −P (v∗ )y
in terms of the orthogonal projectors
P (v) := A(v)A(v)+ ,
⊥
P (v) := Im − P (v) .
In this formulation, the parameters u have been totally eliminated. We might
therefore aim at directly solving the substitute problem [105]
G(v) = min
where G : D ⊆ Rn−s → Rm is defined as
⊥
G(v) := P (v)y .
The associated Gauss-Newton method would then read
Δvk := −G (v k )+ G(vk ) ,
v k+1 := vk + Δvk .
(4.21)
The convergence of this iteration is clearly covered by Theorem 4.3, since the
Moore-Penrose pseudo-inverse is just a special inner inverse; in addition, the
arising projector in the image space of G is orthogonal.
An interesting improvement [128] starts from the differentiation of the projector, which here specifies to
G (v) =
∂ ⊥
P (v)y = −P (v)y .
∂v
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4 Least Squares Problems: Gauss-Newton Methods
⊥
Since G(v) = P (v)G(v), an intriguing idea is to replace
⊥
⊥
G (v) −→ P (v)G (v) = P (v)A (v)A+ (v)y
in iteration (4.21). We then end up with the modified method (k = 0, 1, . . .):
⊥
+
Δv k := − P (vk )G (v k ) G(v k ) ,
vk+1 := vk + Δvk .
(4.22)
Local convergence for this variant, too, is guaranteed, since here also a MoorePenrose pseudo-inverse arises, which includes as well the orthogonality of the
projector in the image space of G.
Numerical realization. Any of the separable Gauss-Newton methods just requires standard QR-decomposition as in the unconstrained case. The simultaneous computation of the iterates vk and uk can be simplified, if the MoorePenrose pseudo-inverse A+ (v) is replaced by some more general inner inverse
A− (v) retaining the property that P (v) := A(v)A− (v) remains an orthogonal
projector.
Positivity constraints. Whenever the amplitudes u = (u1 , . . . us ) must be
positive, then a transformation of the kind ui = û2i appears to be helpful. In
this case, a special Gauss-Newton method for the ûi must be realized aside
instead of just one linear least squares step.
Summarizing, numerical experiments clearly demonstrate that separable
Gauss-Newton methods may well pay off compared to standard GaussNewton methods when applied to separable nonlinear least squares problems.
Bibliographical Note. The first idea about some separable GaussNewton method seems to date back to D. Braess [38] in 1970. His algorithm,
however, turned out to be less efficient than the later suggestions due to
G.H. Golub and V. Pereyra [105] from 1973, which, in turn, was superseded
by the improved variant of L. Kaufman [128] in 1975. Theoretical and numerical comparisons have been performed by A. Ruhe and P.-Å. Wedin [179].
4.2.2 Global Gauss-Newton methods
We now turn to the convergence analysis and algorithm design for the global
Gauss-Newton iteration as defined in (4.13) with damping parameter in the
range 0 < λ ≤ 1. For ease of presentation, we concentrate on the special case
that the projectors P (x) are orthogonal—which is the typical case for any
unconstrained Gauss-Newton method.
As in the preceding theoretical treatment of Newton and Gauss-Newton
methods, we first study the local descent from some xk to the next iterate xk+1 = xk + λΔxk in terms of an appropriate level function and next
prove the global convergence property on this basis. The first theorem is a
confluence of Theorem 3.7 for the global Newton method and Theorem 4.3
for the local Gauss-Newton method.
4.2 Residual Based Algorithms
189
Theorem 4.4 Let F : D ⊂ Rn → Rm with m > n be a differentiable mapping where D ⊂ Rn is open and convex. Let F (x)− denote an inner inverse of
the possibly rank-deficient Jacobian (m, n)-matrix and P (x) := F (x)F (x)−
a projector in the image space of F . Assume the affine contravariant Lipschitz
condition
F (y) − F (x) (y − x) ≤ ωF (x)(y − x)2
for y − x ∈ N ⊥ (F (x)) and x, y ∈ D .
together with
⊥
⊥
P (y)F (y) − P (x)F (x) ≤ ρ(x)F (x)(y − x)
(4.23)
ρ(x) ≤ ρ < 1 for y − x ∈ N ⊥ (F (x)) and x , y ∈ D .
At iterate xk define hk := ωP (xk )F (xk ). Let Lk = x ∈ D| P (x)F (x) ≤
P (xk )F (xk ) ⊂ D denote a compact level set. Then the iterative projected
residuals may be reduced at an estimated rate
P (xk + λΔxk )F (xx + λΔxk ) ≤ tk (λ)P (xk )F (xk ) ,
(4.24)
where
tk (λ) = 1 − (1 − ρ)λ + 12 λ2 hk .
The optimal choice of damping factor in terms of this local estimate is
1−ρ
λk := min 1,
.
(4.25)
hk
Proof. Let xk+1 = xk + λΔxk . For the purpose of estimation we may conveniently start from the identity
⊥
F (xk+1 ) = P (xk )F (xk ) + (1 − λ)P (xk )F (xk )
λ
+
k
F (x + sΔxk ) − F (xk ) Δxk ds .
(4.26)
s=0
From this, we may derive
⊥
P (xk+1 )F (xk+1 ) = F (xk+1 ) − P (xk+1 )F (xk+1 )
⊥
⊥
≤ P (xk+1 )F (xk+1 ) − P (xk )F (xk ) + (1 − λ)P (xk )F (xk )
+
λ
s=0
F (xk + sΔxk ) − F (xk ) Δxk ds .
Note that xk+1 − xk ∈ N ⊥ F (xk ) . Hence, the first right hand term can be
estimated by means of assumption (4.23) to obtain
190
4 Least Squares Problems: Gauss-Newton Methods
⊥
⊥
P (xk+1 )F (xk+1 ) − P (xk )F (xk ) ≤ ρλP (xk )F (xk ) .
The second term is trivial and the third term yields from the Lipschitz condition
λ
F (xk + sΔxk ) − F (xk ) Δxk ds ≤ 12 λ2 hk P (xk )F (xk ) .
s=0
Upon combining these estimates, the result (4.24) is verified. Finally, the
optimal damping factor λ̄k can be seen to minimize the parabola tk (λ) subject
to λ ≤ 1 for the local convergence case.
On the basis of the local Theorem 4.4 we are now ready to derive a global
convergence theorem.
Theorem 4.5 Notation and assumptions as in the preceding Theorem 4.4.
In addition, let D0 ⊆ D denote the path-connected component of the level set
L0 in x0 assumed to be compact. Then the damped Gauss-Newton iteration
(k = 0, 1, . . .) with damping factors in the range
λk ∈ [ε , 2λk − ε]
and sufficiently small ε > 0, which depends on D0 , converges to some solution
point x∗ with P (x∗ )F (x∗ ) = 0.
Proof. The proof is by induction using the local results of the preceding
theorem. It is a direct copy of the proof of Theorem 3.8 for the global Newton method—only replace the residuals by the projected residuals and the
associated level sets.
4.2.3 Adaptive trust region strategy
The clear message from the above global convergence theory is that monotonicity of the iterates should not be required directly in terms of the nonlinear residual norm F (x) but in terms of the projected residual norm
P (x)F (x). Let QR-decomposition be used for the computation of the
(unconstrained) Gauss-Newton corrections. Then, in the notation of Section 4.2.1, (4.18), we have to assure monotonicity as
Θk (λ) =
P (xk + λxk )F (xk + λΔxk )
ck+1 (λ)
=
< 1,
ck P (xk )F (xk )
wherein ck+1 (0) = ck holds. Unfortunately, the monotonicity criterion would
require a Jacobian evaluation also for rejected trial iterates xk + λΔxk . In
4.2 Residual Based Algorithms
191
order to avoid this unwanted computational amount, we suggest to replace
the evaluation of Θk by the much cheaper evaluation of
k
k
k
k (λ) := P (x )F (x + λΔx ) = ĉk+1 (λ) ,
Θ
ck P (xk )F (xk )
(4.27)
where ĉk+1 (0) = ck also holds. From the above Theorem 4.4 we know that
Θk ≤ 1 − (1 − ρ)λ + 12 λ2 hk ,
(4.28)
k can be shown to satisfy
whereas Θ
k ≤ 1 − λ + 1 λ2 hk .
Θ
2
(4.29)
k does not ‘see’ the small residual factor ρ. Therefore,
Colloquially speaking, Θ
0k < 1, we will have to ‘mimic’ Θk < 1 by virtue
rather than just requiring Θ
k .
of some modified test in terms of Θ
Bit counting lemma. Following our paradigm in Section 1.2.3, we replace
the optimal damping factor λk from (4.25) by
1 − [ρ]
λk := min 1,
[hk ]
in terms of computational estimates [ρ] and [hk ] replacing the unavailable
small residual factor ρ and the Kantorovich quantities hk . Then a modified
k can be derived by means of the following lemma.
monotonicity test with Θ
Lemma 4.6 Notation as just introduced. Let ρ = [ρ] < 1 for simplicity. For
some 0 < σ < 1 assume that
0 ≤ hk − [hk ] < σ max (1 − ρ, [hk ]) ESp.
Then for λ = λk the following monotonicity results hold:
Θk ≤ 1 − 12 (1 − ρ)(1 − σ)λ < 1 ,
0k ≤ 1 − 1 − 1 (1 − ρ)(1 + σ) λ < 1 − ρλ .
Θ
2
(4.30)
(4.31)
Proof. We insert the relation
[hk ] ≤ hk < (1 + σ) max(1 − ρ, [hk ])
first into (4.28) to obtain
Θk ≤ 1 − (1 − ρ)λ + 12 λ(1 − ρ)(1 + σ) ,
which verifies (4.30), and second into (4.29) to obtain
k ≤ 1 − λ + 1 λ(1 − ρ)(1 + σ) ,
Θ
2
k .
which leads to (4.31). Finally, insert σ = 1 into Θk and Θ
192
4 Least Squares Problems: Gauss-Newton Methods
Residual monotonicity tests. On this basis, we recommend to mimic the
condition Θk < 1 by the substitute condition
k < 1 − ρλ
Θ
(4.32)
k evaluated as indicated in (4.27). Alternatively, one might also first
with Θ
use condition (4.32) as a-priori monotonicity test and second, after the computation of the new correction, use the condition Θk < 1 in terms of the
above defined contraction factor Θk , as a-posteriori monotonicity test. Furthermore, if we insert σ = 12 in (4.31), then we arrive at the restricted a-priori
monotonicity test
k < 1 − 1 (1 + 3ρ)λ .
Θ
(4.33)
4
In order to design an adaptive trust region strategy, we are only left with
the task of constructing affine contravariant computational estimates [ρ] and
[hk ].
Correction strategy. From (4.26) we may directly obtain
⊥
F (xk +λΔxk )−(1−λ)P (xk )F (xk )−P (xk )F (xk ) ≤ 12 λ2 hk P (xk )F (xk ) .
Note that the left hand term can be easily evaluated using (4.18) and (4.19).
We thus obtain the cheaply computable a-posteriori estimate
[hk ] :=
1/2
2 ck+1 − (1 − λ)ck 2 + dk+1 − dk 2
λ2 ck ≤ hk ,
(4.34)
which can be exploited for the following correction strategy (i = 0, 1, . . .):
1 − [ρ]
i+1
1 i
λk := min 2 λk ,
.
(4.35)
[hi+1
k ]
Prediction strategy. Next, for an a-priori estimate we may simply use
[h0k+1 ] = Θk (λk )[hik∗ ] ,
wherein i∗ indicates the final computed index within estimate (4.34) from
the previous iterative step k. With this estimate we are now able to define
the prediction strategy
1 − [ρ]
λ0k+1 := min 1 ,
,
(4.36)
[h0k+1 ]
4.3 Error Oriented Algorithms
193
Small residual factor estimate. Both of the above strategies require an
estimate of the factor ρ. For this we may just insert available quantities into
the definition (4.23). The newest available information at iterate xk+1 is
⊥
ρk+1
⊥
P (xk+1 )F (xk+1 ) − P (xk )F (xk )
=
≤ ρ.
λk P (xk )F (xk )
The above numerator requires both orthogonal transformations Qk , Qk+1
based on Jacobian evaluations F (xk ), F (xk+1 ). A quite economic version of
evaluating ρk+1 computes first
c̃k
⊥ k
0
k
T
Qk+1 P (x )F (x ) = Qk+1 Qk
=
dk
d˜k
and then
ρk+1 =
c̃k 2 + dk+1 − d˜k 2
λk ck .
Consequently, we will insert
[ρ] = ρk
into (4.35), (4.32) and (4.33)
and
[ρ] = ρk+1
into (4.36) .
Of course, the iteration is terminated whenever ρk ≥ 1 occurs at some iterate
xk . In this case, a ‘large residual’ problem has been identified that cannot be
solved by a Gauss-Newton method.
The adaptive trust region method worked out here on an affine contravariant
basis will be implemented in the code NLSQ-RES.
Bibliographical Note. Most of the published convergence studies about
Gauss-Newton methods for nonlinear least squares problems focus on the
residual convergence aspect, but do not care too much about affine contravariance. The same is true for the implemented algorithms—see, e.g.,
the algorithm by P. Lindström and P.-Å. Wedin [144]. As a matter of fact,
pure Gauss-Newton algorithms are definitely less popular than LevenbergMarquardt algorithms—see, e.g., [118, 152, 54]. Note, however, that when
more than one solution exists, Levenberg-Marquardt methods may supply
‘a numerical solution’ that cannot be interpreted in terms of the underlying
model and data—which is a most undesirable occurrence in scientific computing.
4.3 Error Oriented Algorithms
This section deals with extensions of the error oriented Newton methods for
nonlinear equations (see Section 2.1) to the case of nonlinear least squares
194
4 Least Squares Problems: Gauss-Newton Methods
problems. Attention focuses on a special class of Gauss-Newton algorithms to
be characterized as follows: For given initial guess x0 , let the sequence {xk }
be defined by
Δxk := −F (xk )− F (xk ) , xk+1 = xk + λk Δxk ,
0 < λk ≤ 1 ,
where F : D ⊆ Rn → Rm and m > n. Herein F (x)− denotes an outer inverse
of the Jacobian (m, n)-matrix F (x)—recall property (4.6), which implies that
F (x)− F (x)F (x)− F (x) = F (x)− F (x)
or, in terms of the projectors
P (x) := F (x)− F (x),
P ⊥ = In − P
that
P (x)2 = P (x),
P (x)Δx = Δx .
(4.37)
The presentation covers both local Gauss-Newton methods (λ = 1) and global
Gauss-Newton methods (0 < λ ≤ 1). At first glance, affine covariance in the
domain space does not seem to be a valid concept here; however, as will be
shown below, there still exists a hidden affine covariance structure even in
the case of nonlinear least squares problems, which will be exploited next.
4.3.1 Local convergence results
This section is a direct extension of Section 2.1. We start with a local convergence theorem of Newton-Mysovskikh type, which may be compared with
Theorem 2.2, the affine covariant Newton-Mysovskikh theorem for the ordinary Newton method.
Theorem 4.7 Let F : D ⊆ Rn → Rm with D ∈ Rn open convex denote a
continuously differentiable mapping. Let F (·)− denote an outer inverse of
the Jacobian matrix with possibly deficient rank. Assume that one can find a
starting point x0 ∈ D, a mapping κ : D → R+ and constants α, ω, κ ≥ 0
such that
Δx0 ≤ α ,
F (z)− F (y) − F (x) (y − x) ≤ ωy − x2 ,
for all x, y, z ∈ D collinear, y − x ∈ R F (x)− ,
− ⊥
F (y) P (x)F (x) ≤ κ(x)y − x for all x, y ∈ D ,
κ(x) ≤ κ < 1 for all x ∈ D ,
h := αω < 2(1 − κ) ,
S(x0 , ρ) ⊂ D with ρ := α/(1 − κ − 12 h) .
(4.38)
4.3 Error Oriented Algorithms
195
Then:
(I) The sequence {xk } of local Gauss-Newton iterates (with λk = 1) is welldefined, remains in S(x0 , ρ) and converges to some x∗ ∈ S(x0 , ρ) with
F (x∗ )− F (x∗ ) = 0 .
(II) The convergence rate can be estimated according to
xk+1 − xk ≤ κ(xk−1 ) + 12 ωxk − xk−1 xk − xk−1 .
(4.39)
Proof. Let xk−1 , xk ∈ D for k ≥ 1. Then the following estimates hold:
xk+1 − xk = F (xk )− F (xk )
≤ F (xk )− F (xk ) − F (xk−1 ) − F (xk−1 )(xk − xk−1 ) +F (xk )− I − F (xk−1 )F (xk−1 )− F (xk−1 )
≤ 12 ωxk − xk−1 2 + κ(xk−1 )xk − xk−1 .
This confirms (4.39).
The rest of the induction proof follows along the usual lines of such convergence proofs. Finally, continuity of F (x∗ )− F (xk ) and boundedness of
F (x∗ )− F (xk ) for xk → x∗ help to complete the proof.
Note that the above condition (4.38) may be equivalently written in the form
⊥
⊥
−
(4.40)
F (y) P (y)F (y) − P (x)F (x) ≤ κ(x)y − x ,
which can be more directly compared with the small residual condition (4.14).
The above convergence theorem states existence of a solution x∗ in the sense
that F (x∗ )− F (x∗ ) = 0 even in the situation of deficient Jacobian rank,
which may even vary throughout D as long as ω < ∞ and κ < 1 are assured.
In order to show uniqueness, full rank will be required to extend Theorem
2.3, as will be shown next.
Theorem 4.8 Under the assumptions of Theorem 4.7, let a solution x∗ with
F (x∗ )− F (x∗ ) = 0 exist. Let
σ := x0 − x∗ < σ := 2 1 − κ(x∗ )
ω
and assume full Jacobian rank such that
P (x) = F (x)− F (x) = In for all x ∈ D .
Then the following results hold:
196
4 Least Squares Problems: Gauss-Newton Methods
(I) For any starting point x0 ∈ S(x∗ , σ), the Gauss-Newton iterates remain
in S(x∗ , σ) and converge to x∗ at the estimated rate
xk+1 − x∗ ≤ κ(x∗ ) + 12 ωxk − x∗ xk − x∗ .
(4.41)
(II) The solution x∗ is unique in the open ball S(x∗ , σ).
(III) The following error estimate holds
xk − x∗ ≤
xk+1 − xk 1
− 2 ωxk+1 − xk .
1−κ
(4.42)
Proof. First, we compare the assumptions of Theorem 4.7 and 4.8. With
κ(x∗ ) ≤ κ < 1, one immediately obtains
α<
2
2
(1 − κ) ≤
1 − κ(x∗ ) = σ .
ω
ω
Therefore, the assumption here
x0 − x∗ = σ < σ
is sharper than the corresponding result of Theorem 4.7, since there
x0 − x∗ ≤ ρ =
α
σ
<
.
1 − κ − 12 h
1 − κ − 12 h
On this basis we may therefore proceed to estimate the convergence rate as
xk+1 − x∗ = xk − x∗ − F (xk )− F (xk )
≤ F (xk )− F (x∗ ) − F (xk ) − F (xk )(x∗ − xk ) + F (xk )− F (x∗ )
The first term vanishes as xk −→ x∗ . The same is true for the second term,
since F (x∗ )− F (x∗ ) = 0 implies
F (xk )− F (x∗ ) = F (xk )− (I − F (x∗ )F (x∗ )− )F (x∗ )
≤ κ(x∗ )xk − x∗ .
Thus one confirms (4.41). The proof of contraction follows inductively from
1
ωxk
2
− x∗ + κ(x∗ ) ≤ 12 ωx0 − x∗ + κ(x∗ ) < 12 ωσ + κ(x∗ ) < 1 .
The error estimate (4.42) follows directly by shifting the index from k = 0
to general k within the above assumptions.
Finally, we prove uniqueness by contradiction: let x∗∗ = x∗ denote a different
solution, then (4.41) and the above assumptions with x0 := x∗∗ would imply
that
x∗∗ − x∗ ≤ κ(x∗ ) + 12 ωx∗∗ − x∗ x∗∗ − x∗ < x∗∗ − x∗ .
Hence, x∗∗ = x∗ , which completes the proof.
4.3 Error Oriented Algorithms
197
4.3.2 Local Gauss-Newton algorithms
The above local convergence theorems will now be illustrated for three types
of methods—unconstrained, separable, and constrained Gauss-Newton methods. On this basis, details of the realization of the algorithms will be worked
out.
Unconstrained Gauss-Newton method. Consider once again the unconstrained nonlinear least squares problem (4.2). The corresponding GaussNewton method is then
Δxk := −F (xk )+ F (xk ) , xk+1 = xk + Δxk ,
(4.43)
with the specification of the Moore-Penrose pseudo-inverse:
F (x)− = F (x)+ .
At the solution point x∗ , the condition
F (x∗ )+ F (x∗ ) = 0
will hold, which is equivalent to
F (x∗ )T F (x∗ ) = 0 ,
since in finite dimensions
N F (x)T = N F (x)+ .
Numerical realization. Each iterative step of the Gauss-Newton method (4.43)
simply realizes any of the numerical techniques for linear least squares
problems—such as the QR-decomposition (4.7) described in Section 4.1.1,
which, in the realistic case m >> n, requires a computational amount of
∼ 2mn2 .
Warning. We want to mention explicitly that any ‘simplified’ Gauss-Newton
method with a specification of the kind
F (x)− = F (x0 )+
will usually converge to some point x̂ = x∗ depending on x0 and therefore
solve the ‘wrong’ nonlinear least squares problem
F (x0 )+ F (x̂) = 0 .
(4.44)
A similar warning applies to most ‘Gauss-Newton-like’ or ‘quasi-GaussNewton’ methods (as long as m > n). In the optimization literature such
an occurrence is well-known as ‘caging’ of the iterates depending on the initial guess.
198
4 Least Squares Problems: Gauss-Newton Methods
Incompatibility factor. Upon comparing the convergence theorems for
Newton and Gauss-Newton methods, the essential new item seems to be the
condition κ(x) < 1, which restricts the class of problems that are successfully
tractable by a local Gauss-Newton method. In order to gain some insight into
this condition, it is now studied for several cases.
(I) In purely linear least squares problems, one has
F (x)− = F (x)+ , F (x) = F (y) ,
which directly implies the best possible choices
ω = 0 , κ(x) ≡ κ = 0 .
As a consequence, one iteration step produces x∗ = x1 independent of x0 .
(II) For general systems of nonlinear equations (m = n), let F (x) be nonsingular for all x ∈ D. Then
F (x)− = F (x)−1 ⇒ κ(x) ≡ 0 ,
which guarantees the usual quadratic convergence of the ordinary Newton
method.
(III) For general nonlinear least squares problems (m > n) Theorem 4.7,
which is independent of the actual Jacobian rank, states that
xk+1 − xk ≤ κ(x∗ ) ,
k→∞ xk − xk−1 lim
showing that κ(x∗ ) is the asymptotic convergence rate.
(IV) An interesting reformulation of condition (4.38) at x = x∗ is
κ(x∗ ) := sup
x∈D
F (x)+ F (x∗ )
< 1.
x − x∗ Obviously, this condition rules out problems with ‘too large’ residual F (x∗ )
in terms of second order derivative information.
For compatible problems especially, the definition of κ directly implies that
κ(x∗ ) = 0 .
In this interpretation κ measures the incompatibility and is therefore called
incompatibility factor herein. Rather than speaking of ‘small residual’ problems, we want to coin a different wording here:
Definition: Adequate nonlinear least squares problems are characterized by
the condition
κ(x∗ ) < 1 .
4.3 Error Oriented Algorithms
199
Assume that condition (4.38) permits the definition of a continuous mapping
κ in a neighborhood D0 ⊆ D of x∗ , where we assume D0 to be restricted
such that κ(x) ≤ κ < 1. In short, Theorem 4.7 may then be summarized as:
The Gauss-Newton method converges locally linearly for adequate and quadratically for compatible nonlinear least squares problems.
Note that a necessary condition for the continuity of κ is that F (x) has
constant rank in D0 , which need not be full rank! If F (x) has full rank, then
P (x) = In guarantees even local uniqueness of the solution x∗ .
Convergence monitor. The convergence of the local Gauss-Newton method
will be monitored by the contraction condition
Θk :=
Δxk+1 <1
Δxk in terms of the Gauss-Newton corrections. From Theorem 4.7 we know that
Θk ≤ κ(xk ) + 12 hk ,
where the hk = ωΔxk denote the associated Kantorovich quantities. In
order to separate the effect of the two right hand terms above, it is advisable
to additionally compute the simplified Gauss-Newton corrections
Δx
k+1
= −F (xk )+ F (xk+1 ) ,
which can be easily shown to satisfy
k+1
Θk :=
Δx ≤ 12 hk .
Δxk For compatible nonlinear least squares problems, the behavior of the two contraction factors will be roughly the same, whereas for incompatible problems
a rather different behavior will show up: as soon as
hk κ ,
we will observe that
k+1
Δxk+1 ≈ κΔxk , Δx
k
≈ κ2 Δx .
(4.45)
This situation is illustrated schematically in Figure 4.1.
Termination criteria. On the basis of the statistical interpretation of the
Gauss-Newton method, we suggest to terminate the iteration as soon as linear convergence dominates the iteration process: iteration beyond that point
would just lead to an accuracy far below reasonable in view of model and
data accuracy. A convenient criterion to detect this point will be
200
4 Least Squares Problems: Gauss-Newton Methods
Θk Θk .
As an alternative, we also use the modified termination criterion
k+1
Δx
≤ XTOL
(4.46)
in terms of some user prescribed error tolerance XTOL. After termination in
the linear convergence phase, the finally achieved accuracy will be
xk+1 − x∗ ≈ k+1 :=
Θk
Δxk .
1 − Θk
(4.47)
Herein the norms are understood to be scaled norms, of course, to permit
such an interpretation. The above criteria also prevent codes from getting
inefficient, if unaware users require a too stringent error tolerance parameter
XTOL. A feature of this kind is important within any nonlinear least squares
algorithm, since already in the purely linear case iterative refinement may fail
to converge—compare the detailed rounding error analysis of G.H. Golub and
J.H. Wilkinson [108] and of Å. Bjørck [25].
1
10−2
10−4
10−6
10−8
10−10
10−12
10−14
10−16
k
0
2
4
6
8
10
12
14
16
18
20
22
24
26
Fig. 4.1. Enzyme reaction problem. Linear convergence pattern of ordinary
Gauss-Newton corrections Δxk 2 (•) and of simplified Gauss-Newton corrections
k+1 2
Δx
(◦) for k ≥ 12.
Example 4.1 Enzyme reaction problem. In order to illustrate the convergence pattern for incompatible problems, the following unconstrained problem due to J. Kowalik and M.R. Osborne [136] is selected (m = 11, n = 4):
11
F (x)2 =
i=1
2
yi − ϕ(ti , x) = min .
4.3 Error Oriented Algorithms
201
With starting point x0 = (0.25, 0.39, 0.415, 0.39), the local Gauss-Newton
method presented here would fail to converge, whereas the global GaussNewton method to be presented in Section 4.3.4 converged after k = 16
iterations with an estimated contraction factor Θ16 ≈ κ ≈ 0.65. The iterative
behavior of the ordinary and the simplified Gauss-Newton corrections is represented graphically in Figure 4.1. In the linear convergence phase a scissors
is seen to open between the iterative behavior of the ordinary and the simplified corrections—just as described in (4.45). In this case, the Gauss-Newton
iteration has been deliberately continued below any reasonable required accuracy. For XTOL = 10−7 the termination criterion (4.46) would have been
activated at k = 16. According to (4.47) this would have led to a finally
achieved accuracy
x17 − x∗ ≈ 17 = 0.9 · 10−2 .
The final value of the least squares objective function appeared to be
F (x∗ )2 = 3.1045 · 10−4 .
A-posteriori perturbation analysis. The above incompatibility factor
still has a further interpretation, this time in terms of statistics. In such problems, the minimization formulation is only reasonable, if small perturbations
of the measured data lead to acceptable perturbations of the parameters,
i.e., if the solution x∗ is stable under small perturbations. Let δF (x∗ ) denote
such a perturbation. Then standard linear perturbation analysis yields the
parameter perturbations
δx∗L := −F (x∗ )+ δF (x∗ ) .
For a nonlinear perturbation analysis, Theorem 4.7 can be applied with
x0 := x∗old ,
x∗new := x∗old + δx∗NL
to obtain the more appropriate estimate
˙
δx∗NL ≤
δx∗L .
1 − κ(x∗ )
As a consequence, stability of the underlying statistical model can only
be guaranteed for adequate least squares problems. In [32] H.G. Bock even
proved that for κ(x∗ ) ≥ 1 the point x∗ is no longer a local minimum of
F (x). This insight leads to the following rules:
Whenever the ordinary Gauss-Newton method with full rank Jacobian fails to
converge, then one should rather improve the model than just turn to a different iterative solver. In the convergent, but rank-deficient case, one should
try to get more information by acquiring better data.
202
4 Least Squares Problems: Gauss-Newton Methods
Bibliographical Note. These rules are backed by careful theoretical considerations in terms of the underlying statistics—for details see, e.g., the early
paper [123] by R.L. Jennrich and the more recent, computationally oriented
study [164] by M.R. Osborne. In the statistics community the Gauss–Newton
method is usually named as scoring method.
Separable Gauss-Newton methods. Separable nonlinear least squares
problems have been defined in (4.20) above as
F (u, v) = A(v)u − y = min ,
wherein the matrix
A(v) : (m, s)-matrix, u ∈ Rs , y ∈ Rm , v ∈ Rn−s
carries the nonlinearity of the model parameters.
Following G.H. Golub and V. Pereyra [105], one aims at solving the substitute problem
⊥
G(v) = P (v)y = min
in terms of the projectors
⊥
P (v) = A(v)A(v)+ , P (v) := Im − P (v) .
The convergence of the associated Gauss-Newton method is covered by Theorem 4.7, since the Moore-Penrose pseudo-inverse is just a special outer inverse.
The simpler variant due to L. Kaufman [128] is characterized by
⊥
+
G (v)− = P (v)G (v) ,
which obviously is again a special outer inverse so that local convergence
for this variant is also guaranteed by Theorem 4.7. There is some theoretical
evidence that the Kaufman variant, which is anyway cheaper to implement,
may also have a possibly larger convergence domain than the Golub-Pereyra
suggestion—compare Exercise 4.8.
Constrained Gauss-Newton method. Consider the nonlinear least squares
problem
G(x)2 = min ,
(4.48)
G : D 2 ⊆ Rn → Rm 2
subject to the nonlinear equality constraints
H(x) = 0 ,
H : D1 ⊂ Rn → Rm1 ,
(4.49)
4.3 Error Oriented Algorithms
203
where D := D1 ∩ D2 = ∅, and
m1 < n < m1 + m2 =: m .
In order to treat the truly nonlinear case (in both the least squares functional
and the constraints), we apply the same kind of idea as for the linear case
in Section 4.1.2. Instead of (4.48) subject to (4.49) we consider the unconstrained penalty problem
G(x)2 + μ2 H(x)2 = min .
With the notation
Fμ (x) :=
μH(x)
G(x)
,
Fμ (x)
:=
μH (x)
G (x)
the above problem is equivalent to
Fμ (x)2 = min
and, for μ < ∞, the associated local Gauss-Newton method is defined via
the corrections
Δxkμ := Fμ (xk )+ Fμ (xk ) .
As in the linear case, the penalty limit μ → ∞ is of interest.
Lemma 4.9 Notation as just introduced. Let
rank H (x) = m1 , x ∈ D ⊆ Rn .
Then, for x, y ∈ D, the following forms exist:
Δ(x, y)
:= − lim Fμ (x)+ Fμ (y) ,
(4.50)
P (x, y)
:=
lim Fμ (x)+ Fμ (y) ,
(4.51)
μ→∞
μ→∞
which satisfy the projection properties:
P T (x, x) = P 2 (x, x) = P (x, x) ,
P (x, x)Δ(x, x) = Δ(x, x) .
Proof. Columnwise application of Lemma 4.2 for the linear case.
In the just introduced notation, the local Gauss-Newton method for the constrained problem—in short: constrained Gauss-Newton method—can be written as:
xk+1 := xk + Δ(xk , xk ) k = 0, 1, . . . .
(4.52)
The convergence of this iteration is easily seen by simply rewriting Theorem 4.7 in terms of the forms P and Δ, which leads to the following local
convergence theorem.
204
4 Least Squares Problems: Gauss-Newton Methods
Theorem 4.10 Notation as just introduced in this section. Let D ⊆ Rn
denote some open convex domain of the differentiable mappings G, H. Let
P (· , ·) , Δ(· , ·) denote the forms introduced in Lemma 4.9. Assume that
one can find a starting point x0 ∈ D, a mapping κ : D → R+ and constants
α, ω, κ ≥ 0 such that
Δ(x0 , x0 ) ≤ α ,
(P (z, y) − P (z, x)) (y − x) ≤ ωy − x2
for all x, y, z ∈ D collinear, y − x ∈ R P (x, x) ,
Δ(y, x) − P (y, x)Δ(x, x) ≤ κ(x)y − x for all x, y ∈ D ,
κ(x) ≤ κ < 1 for all x ∈ D ,
h := αω < 2(1 − κ) ,
S(x0 , ρ) ⊂ D with ρ := α/(1 − κ − 12 h) .
Then:
(I) The sequence {xk } of constrained Gauss-Newton iterates (4.52) is welldefined, remains in S(x0 , ρ) and converges to some x∗ ∈ S(x0 , ρ) with
Δ(x∗ , x∗ ) = 0 .
(II) The convergence rate can be estimated according to
xk+1 − xk ≤ κ(xk−1 ) + 12 ωxk − xk−1 xk − xk−1 .
Numerical realization. Each iterative step of the constrained Gauss-Newton
method (4.52) realizes the penalty limit method described in Section 4.1.2—
or, of course, any other numerical technique for constrained linear least
squares problems (see, e.g., the textbook [107]).
Incompatibility factor. As in the unconstrained case, this factor has an interpretation as asymptotic convergence factor—see Theorem 4.10. Also its statistical interpretation carries over—with the natural modification that now
statistical perturbations are only allowed to produce some δG = 0, but must
preserve the equality constraints so that δH = 0. Similar statements hold for
the a-posteriori perturbation analysis.
Convergence criteria. The monitoring of convergence as well as the termination criterion carry over—just replace the unconstrained corrections by the
appropriate constrained Δ(·, ·).
Inequality constraints. In this frequently occurring case a modification
of this equality constrained Gauss-Newton method is possible. The essential additionally required technique is a so-called active set strategy: active
4.3 Error Oriented Algorithms
205
inequality constraints are included into the set of equality constraints. The
detection criterion for an active inequality is usually based on the sign of the
corresponding Lagrange multiplier. For this reason the decomposition technique due to J. Stoer [186] may be preferable here, which permits an easy
and cheap evaluation of the Lagrange multipliers. Whenever the set of active
inequalities remains constant for x ∈ D, then the above convergence results
apply directly. If, however, the active set changes from iterate to iterate, then
even local convergence cannot be guaranteed. For practical purposes, such
a situation is usually avoided by deliberate suppression of active set loops.
However, proceeding like that might sometimes lead to solving the ‘wrong’
problem.
Bibliographical Note. A first error oriented convergence theorem for
rather general Gauss-Newton methods has been given in 1978 by P. Deuflhard
and G. Heindl [76, Theorem 4]. In parallel, the application to equality constrained Gauss-Newton methods via the penalty limit has been worked out by
P. Deuflhard and V. Apostolescu [69]. Since 1981, theorems of this kind have
also been derived and used for the construction of algorithms for parameter
identification in ODEs by H.G. Bock [29, 31, 32].
4.3.3 Global convergence results
In this section, we will study global convergence of the damped Gauss-Newton
method in the error oriented framework. In order to include both constrained
and unconstrained nonlinear least squares problems in a common treatment,
we will use the forms P (·, ·) and Δ(·, ·) introduced in Lemma 4.9 for both
cases, meaning the limiting definitions (4.51) and (4.50) in the constrained
case, but the definitions
P (x, y) := F (x)+ F (y) , Δ(x, y) := −F (x)+ F (y)
in the unconstrained case. For simplicity, all derivations will be made in the
unconstrained framework.
Global versus local Gauss-Newton path. At first glance, (unconstrained) nonlinear least squares problems do not seem to exhibit any affine
covariance property apart from the trivial class O(m) containing the orthogonal (m, m)-matrices. However, upon careful examination a nontrivial
invariance class for this kind of problem can be detected, which is usually
overlooked in theoretical treatments. For this purpose, just observe that the
nonlinear least squares problem is equivalent to the system of n nonlinear
equations
F (x)+ F (x) = 0 .
Application of Newton’s method to this formulation, however, would require
second-order tensor information beyond mere Jacobian information—which
206
4 Least Squares Problems: Gauss-Newton Methods
is of no interest in the present context. Rather, we will restrict our attention
to the following more special system of equations
F (x∗ )+ F (x) = 0 ,
(4.53)
wherein the Jacobian is fixed at the solution point. This formulation immediately gives rise to the affine covariance class
A(x∗ ) := A = BF (x∗ )+ |B ∈ GL(n) ,
which means that equation (4.53) is equivalent to any of the equations
AF (x) = 0 with
A ∈ A(x∗ ) .
On this basis, we may monitor the Gauss-Newton iteration by means of any
of the test functions
T (x|A) := 12 AF (x)2 ,
A ∈ A(x∗ ) .
The case A = I has been discussed in the preceding Section 4.2 in the affine
contravariant setting. Here we are interested in an extension to the affine
covariant setting. As in the derivation of global Newton methods in Section
3.1.4, we define the level sets
G(z|A) := {x ∈ D|T (x|A) ≤ T (z|A)}
and study their intersection
G∗ (x0 ) :=
G(x0 |A)
(4.54)
A∈A(x∗ )
for a given starting point x0 .
Theorem 4.11 Let F ∈ C 1 (D), D ⊆ Rn and P (x∗ , x) nonsingular for all
∈ A(x∗ ) let the path-connected component D0 of G(x0 |A)
x ∈ D. For some A
0
in x be compact and D0 ⊆ D. Then the path-connected component of G∗ (x0 )
in (4.54) defines a topological path x∗ : [0, 2] −→ Rn , which satisfies:
Δ x∗ , x∗ (λ) = (1 − λ)Δ(x∗ , x0 ) ,
(4.55)
= (1 − λ)2 T (x0 |A)
,
T (x∗ (λ)|A)
dx∗
= Δ(x∗ , x0 ) ,
dλ
x∗ (0) = x0 , x∗ (1) = x∗ ,
P (x∗ , x∗ )
dx∗ = P (x∗ , x0 )−1 Δ(x∗ , x0 ) .
dλ λ=0
(4.56)
4.3 Error Oriented Algorithms
207
Proof. One proceeds as in the proof of Theorem 3.6, but here for the mapping
F (x∗ )+ F . Intersection of level sets in the image space of that mapping leads
to the homotopy
Φ(x, λ) := F (x∗ )+ F (x) − (1 − λ)F (x∗ )+ F (x0 ) = 0
for λ ∈ [0, 2]. One easily verifies that
Φx = F (x∗ )+ F (x) = P (x∗ , x) , Φλ = F (x∗ )+ F (x0 ) = −Δ(x∗ , x0 ) .
The process of local continuation to construct the path x∗ will naturally
start
at x∗ (0) = x0 . As G∗ (x0 ) ⊂ D0 ⊆ D, the nonsingularity of P x∗ , x∗ (λ) is
assured, which permits continuation up to x∗ (1) = x∗ and x∗ (2). From this,
the results (4.55) up to (4.56) follow directly.
∗
Remark 4.2 The somewhat obscure
assumption ‘P (x , x) nonsingular’
cannot be replaced by just ‘rank F (x) = n’. Under the latter assumption,
one concludes from
N P (x∗ , x) = N F (x∗ )+ ∩ R F (x)
that
dim N F (x∗ )+
= m − n,
dim R F (x)
= n.
Hence, even though the generic case will be that
dim N P (x∗ , x) = 0 ,
the case
dim N P (x∗ , x) > 0
cannot be generally excluded. On the other hand, if the Jacobian F (x) is
rank-deficient, then P (x∗ , x) is certainly singular, which violates the corresponding assumption.
The crucial result of the above theorem is that the local tangent direction
of x∗ in x0 requires global information via F (x∗ )—which means that the
tangent cannot be realized within any Gauss-Newton method. If we replace
A(x∗ ) −→ A(x0 ) ,
we end up with a result similar to the one before.
Theorem 4.12 Under assumptions that are the natural modifications of
those in Theorem 4.11, the intersection
208
4 Least Squares Problems: Gauss-Newton Methods
G0 (x0 ) :=
G(x0 |A)
A∈A(x0 )
defines a topological path x0 : [0, 2] −→ Rn , which satisfies
Δ x0 , x0 (λ) = (1 − λ)Δ(x0 , x0 ) ,
dx0
= Δ(x0 , x0 ) ,
dλ
x0 (1) = x∗ in general,
dx0 = Δ(x0 , x0 ) .
dλ λ=0
P (x0 , x0 )
x0 (0) = x0 ,
The proof can be omitted. The most important result is that the tangent
direction of the path x0 in x0 is now the Gauss-Newton direction. Comparing
the two theorems, we will call the path x∗ global Gauss-Newton path and the
path x0 local Gauss-Newton path. The situation is represented graphically in
Figure 4.2. Each new iterate induces a new ‘wrong’ problem
Δ(x0 , x) = 0
via the new local Gauss-Newton path—compare also (4.44). The whole procedure may nevertheless approach the ‘true’ problem under the natural assumption that the tangent directions ẋ∗ (0) and ẋ0 (0) do not differ ‘too much’—a
more precise definition of this term will be given below.
ẋ∗ (0)
ẋ0 (0)
x0
x∗
x0
x∗ = x∗ (1)
x0 (1)
Fig. 4.2. Geometrical scheme: local and global Gauss-Newton path.
Remark 4.3 For m = n = rank(F (x)), which is the nonlinear equation
case, the two paths turn out to be identical—the Newton path treated in
Section 3.1.4.
4.3 Error Oriented Algorithms
209
Convergence analysis. On the basis of the geometrical concept of the
Gauss-Newton paths, we are now ready to study the convergence of the global
Gauss-Newton method including damping.
Theorem 4.13 Notation and assumptions as before. Let the damped GaussNewton iteration be defined as
xk+1 := xk + λΔ(xk , xk ) .
Let x, y, xk , xk+1 ∈ D0 , for some convex D0 ⊆ D. Define a Lipschitz constant
ω∗ by
P (x∗ , y) − P (x∗ , x) (y − x) ≤ ω∗ y − x2 .
Moreover, assume that
Δ(x∗ , x) − P (x∗ , x)Δ(x, x) ≤ δ∗ (x)Δ(x∗ , x) , δ∗ (x) ≤ δ < 1 .
Then, with the convenient notation
h∗k := ω∗ 1 + δ∗ (xk ) P (x∗ , xk )−1 · Δ(xk , xk ) ,
we obtain
Δ x∗ , xk + λΔ(xk , xk ) ≤ t∗k (λ)Δ(x∗ , xk ) ,
wherein
(4.57)
(4.58)
(4.59)
t∗k (λ) := 1 − 1 − δ∗ (xk ) λ + 12 λ2 h∗k .
The globally optimal damping factor is
λ∗k := min 1 , (1 − δ∗ (xk ))/h∗k .
(4.60)
Proof. Using standard tools we arrive at the following estimate:
∗ +
F (x ) F (x
k+1
λ
∗ +
) = F (x )
k
F (x + sΔxk ) − F (xk ) Δxk ds
s=0
+(1 − λ)F (x∗ )+ F (xk ) + λF (x∗ )+ I − F (xk )F (xk )+ F (xk ) .
Assumption (4.57) directly leads to
Δ x∗ , xk + λΔ(xk , xk ) ≤ 1 − λ 1 − δ∗ (xk ) Δ(x∗ , xk ) + 12 λ2 ω∗ Δ(xk , xk )2 .
The second order term needs additional treatment:
Δ(xk , xk ) ≤ P (x∗ , xk )−1 · P (x∗ , xk )Δ(xk , xk )
≤ P (x∗ , xk )−1 1 + δ∗ (xk ) Δ(x∗ , xk ) .
210
4 Least Squares Problems: Gauss-Newton Methods
Upon combining these estimates and using the notation for h∗k , one obtains
the result (4.59) and the optimal damping factor (4.60).
At iterate xk , the above assumption (4.57) is equivalent to
P (x∗ , xk ) ẋ∗ (0) − ẋk (0) ≤ δ∗ (xk ) ≤ δ̄ < 1 ,
P (x∗ , xk )ẋ∗ (0)
which has a nice geometric interpretation in terms of Figure 4.2: in this
form, the condition is seen to restrict the ‘directional discrepancy’ of the
local Gauss-Newton path xk and the global Gauss-Newton path x∗ starting
at xk . The assumption may also be characterized in algebraic terms (for the
unconstrained case):
⊥
F (x∗ )+ P (x)F (x) ≤ δ∗ (x)F (x∗ )+ F (x) .
(4.61)
With these preparations, we are now ready to state global convergence.
Theorem 4.14 Notations and assumptions as before in Theorem 4.13. Let
P (x, x) = I and P (x∗ , x) be nonsingular
for all x ∈ D. Denote the pathconnected component of G x0 |F (x∗ )+ in x0 by D0 and assume D0 ⊆ D,
D0 compact. Then the Gauss-Newton iteration with damping factors
λk ∈ [ε , 2λ∗k − ε]
converges globally to x∗ . In addition, there exists an index k0 ≥ 0 such that
λ∗k = 1 for all k ≥ k0 .
In comparison with Theorem 4.7 the asymptotic convergence factors show the
relation
lim∗ δ∗ (x) = κ(x∗ )
(4.62)
x→x
assuming best possible estimates for δ∗ (x) and κ(x).
Proof. The first part of the proof is just along the lines of earlier proofs
of global convergence made so far—and is therefore omitted. A new piece
of proof comes up with the verification of the above merging property. For
this purpose we merely need to study the limit for xk → x∗ , since global
convergence is already guaranteed. For best possible estimates and D0 →
Dk → D∗ = {x∗ } one obtains
δ∗ (x)
−→
δ∗ (x∗ ) ≤ δ < 1 ,
P (x∗ , xk )
−→
P (x∗ , x∗ ) = In ,
P (x∗ , xk )−1 −→
1,
h∗k
−→
0.
4.3 Error Oriented Algorithms
211
Hence, if xk is close enough to x∗ , then h∗k decreases monotonically.
Finally, the connection property (4.62) can be shown rather elementarily.
Just recall condition (4.40) with y = x∗ and use Taylor’s expansion for the
right hand side in (4.61) to obtain
F (x∗ )+ (F (x∗ ) + F (x∗ )(x − x∗ ) + · · · ) = P (x∗ , x∗ )(x − x∗ ) + O(x − x∗ 2 ) .
Hence, with P (x∗ , x∗ ) = I and xk → x∗ , we have verified (4.62), which
completes the proof.
Instead of the unavailable solution point x∗ we will certainly insert the best
available iterate xk . This leads to the following theorem.
Theorem 4.15 Notation and assumptions as in Theorem 4.13 before. Let
x, y, xk , xk+1 ∈ D0 ⊆ D. Define Lipschitz constants ωk by
P (xk , y) − P (xk , x) (y − x) ≤ ωk y − x2 .
(4.63)
Then, with the notation
hk := ωk Δ(xk , xk )| ,
we obtain
k k
Δ x , x + λΔ(xk , xk ) ≤ 1 − λ + 1 λ2 hk Δ(xk , xk ) .
2
(4.64)
Compared with (4.58) we obtain
hk ≤ h∗k .
(4.65)
Proof. As in the derivations before, we may obtain the following estimate:
F (xk )+ F (xk+1 ) =
λ
F (x )
k +
F (xk + sΔxk ) − F (xk ) Δxk ds + (1 − λ)F (xk )+ F (xk ) ,
s=0
which yields (4.64). Finally, using best possible estimates for ωk , ω∗ as defined
in the two above theorems, one may verify that
ωk ≤ ω∗ P (x∗ , xk )−1 1 + δ∗ (xk ) ,
which implies (4.65).
212
4 Least Squares Problems: Gauss-Newton Methods
4.3.4 Adaptive trust region strategies
As for the global convergence of the damped Gauss-Newton method, the situation can be described as follows: in order to guarantee global convergence,
global Jacobian information at x∗ would be required. Since such information
is computationally unavailable, all global information must be replaced by local information at the newest iterate xk —at the expense that then there is no
guarantee of convergence. For the time being, assume full rank P (x, x) = In
for all arguments x in question.
Natural level function. First, the global level function
T x|F (x∗ )+ = 12 Δ(x∗ , x)2
is replaced by its local counterpart
T x|F (xk )+ = 12 Δ(xk , x)2 ,
which is called natural level function in view of the detailed investigations
for nonlinear equations in Section 3.3.2. Even in the extended case here, this
level function has intriguing properties:
(I) Application of the Penrose axioms directly leads one to
Δ(xk , xk ) = − grad T x|F (xk )+ x=xk ,
for arbitrary Jacobian rank. This means that the Gauss-Newton direction is the direction of steepest descent with respect to the natural level
function.
(II) For F ∈ C 2 (D), one easily verifies that
Δ(x∗ , x) = P (x∗ , x∗ )(x − x∗ ) + O x − x∗ 2 ,
which then implies
T x|F (x∗ )+ = 12 x − x∗ 2 + O(x − x∗ 3 ) .
Obviously, for xk → x∗ , the natural level function is an asymptotic distance function.
Convergence criteria. As a consequence of Theorem 4.13, the iterates
might be tested via the contraction factor
Θk∗ :=
Δ(x∗ , xk+1 ) ≤ 1 − (1 − δ)λ + 12 λ2 h∗k ,
∗
k
Δ(x , x )
which, however, contains computationally unavailable information. Therefore
we again suggest to ‘mimic’ the situation by means of a substitute test based
4.3 Error Oriented Algorithms
213
on Theorem 4.15. We thus arrive at the computationally available contraction
factor
Δ(xk , xk+1 ) Θk :=
≤ 1 − λ + 12 λ2 hk
Δ(xk , xk )
containing the simplified Gauss-Newton correction, which in the unconstrained case reads
Δ(xk , xk+1 ) = −F (xk )+ F (xk+1 ) .
Colloquially speaking, the contraction factor Θk does not ‘see’ the incompatibility factor κ.
Bit counting lemma. In order to derive the desired substitute test, we may
exploit the following lemma.
Lemma 4.16 Notation as just introduced. Let δ = [δ] < 1 for simplicity.
For some 0 < σ < 1 assume that
0 ≤ hk − [hk ] < σ max 1 − δ, [hk ] .
Then for the damping factor
1−δ
λ = λk = min 1,
[hk ]
the following monotonicity results hold:
Θk∗ ≤ 1 − 12 (1 − δ)(1 − σ)λ < 1 ,
(4.66)
Θk ≤ 1 − 1 − 12 (1 − δ)(1 + σ) λ < 1 − δλ .
(4.67)
Proof. As in the proof of Lemma 4.6 we just insert the relation
[hk ] ≤ hk < (1 + σ) max(1 − δ, [hk ])
to obtain (4.66) and (4.67) and set σ = 1.
Natural monotonicity test. On this basis, we now recommend to replace
the condition Θk∗ < 1 by the test
Θk < 1 − δλ .
If we insert σ =
1
2
in (4.67), then we arrive at its restricted counterpart
Θk < 1 − 14 (1 + 3δ)λ .
214
4 Least Squares Problems: Gauss-Newton Methods
Correction strategy. In order to design an adaptive trust region strategy,
we are left with the task of constructing cheap computational estimates [δ]
and [hk ]. Note that due to (4.65) the estimates [hk ] will automatically be also
estimates [h∗k ]. From Theorem 4.14 the a-posteriori estimate
2 Δ xk , xk + λΔ(xk , xk ) − (1 − λ)Δ(xk , xk ) [hk ] :=
≤ hk
(4.68)
λ2 Δ(xk , xk )
can be obtained, which supplies the correction strategy (i = 0, 1, . . .):
1−δ
1
λi+1
:=
min
λ,
.
(4.69)
k
2
[hk (λ)] λ=λik
Prediction strategy. For its derivation we will proceed as in the nonlinear
equation case and slightly modify the Lipschitz condition (4.63). Let some
Lipschitz constant ω k be defined by
P (xk , y) − P (xk , x) v ≤ ω k y − xv .
By construction, we have ω k ≥ ωk . Upon specification of the above arguments, we may exploit the relation
P (xk , xk ) − P (xk , xk−1 ) Δ(xk−1 , xk ) ≤ ω k xk − xk−1 Δ(xk−1 , xk )
to obtain the a-priori estimate
[ω k ] :=
Δ(xk−1 , xk ) − Δ(xk , xk ) + Δ(xk , xk−1 )
≤ ωk ,
λk−1 Δ(xk−1 , xk−1 )Δ(xk−1 , xk )
(4.70)
wherein in the unconstrained case
Δ(xk , xk−1 ) := −F (xk )+ F (xk ) + F (xk−1 )Δ(xk−1 , xk ) .
This establishes the prediction strategy for k > 0 as
1−δ
0
λk := min 1,
.
[ωk ] Δ(xk , xk )
(4.71)
Finally, we are still left with the specification of the quantity δ in the choice of
the damping factors. Once again, we will replace the condition δ∗ (x) ≤ δ < 1
from (4.57) by some substitute local condition
δk (x) :=
Δ(xk , x) − P (xk , x)Δ(x, x)
≤δ<1
Δ(xk , x)
for appropriate arguments x. For x = xk , the Penrose axioms trivially yield
δk (xk ) = 0. For x = xk−1 , k > 0 the condition δk (xk−1 ) < 1 may naturally
4.3 Error Oriented Algorithms
215
be imposed. Hence, either [δ] := 0 or [δ] = δk (xk−1 ) might be used in actual
computation. As soon as the iterates approach the solution point (i.e. at least
λk = 1), we switch to the local option
[δ] =
Δ(xk+1 , xk+1 )
≈ κ(xk )
Δ(xk , xk )
based on the local contraction result (4.39) and the asymptotic result (4.62).
Remark 4.4 For inadequate nonlinear least squares problems, the adaptive
damping strategy will typically supply values
λk ≈ 1/κ < 1 .
Vice versa, this effect can be conveniently taken as an indication of the inadequacy of an inverse problem under consideration.
In most realizations of global Gauss-Newton methods, the choice δ k := 0 is
made ad-hoc. As in the simpler Newton case, numerical experience has shown
that mostly λ0k is successful, whereas λ1k has turned out to be sufficient in
nearly all remaining cases. The whole strategy appeared to be extremely
efficient even in sensitive nonlinear least squares problems up to large scale—
despite the lack of a really satisfactory global convergence proof.
4.3.5 Adaptive rank strategies
In contrast to the preceding section, the full rank assumption for P (x, x) is
now dropped. For ease of writing, we will introduce the short hand notation J(x) := F (x). In the unconstrained case, the situation of possible rank
deficiency
q(x) := rank J(x) ≤ n , P (x, x) = In for q < n
may arise either if the Jacobian at some iterate turns out to be ill-conditioned
or if a deliberate rank reduction has been performed, say, by replacing
J(x) −→ J (x) with q := rank J (x) < q .
In the rank-deficient case, the corrections Δ(x0 , x0 ) are confined to the cokernel of J(x0 ), since
P (x0 , x0 )Δ(x0 , x0 ) = Δ(x0 , x0 ) .
As an immediate consequence, the Gauss-Newton method starting at x0 is
locally solving either the equations
J(x0 )+ F x0 + P (x0 , x0 )z = 0
216
4 Least Squares Problems: Gauss-Newton Methods
or the equations
J (x0 )+ F x0 + P (x0 , x0 )z = 0 ,
where the intuitive notation P (x0 , x0 ) just denotes the associated rankdeficient projector. On this basis, a Gauss-Newton method with damping
strategy may be constructed in the rank-deficient case as well. In order to
do so, the following feature of the associated Lipschitz constants is of crucial
importance.
Lemma 4.17 Associated with the Jacobian matrices J(x0 ) and J (x0 ), respectively, let ω0 ≤ ω0 , ω0 ≤ ω 0 denote the best possible Lipschitz constants
in the notation of Section 4.3.4. Then rank reduction implies that
ω0 ≤ ω0 , ω 0 ≤ ω 0 .
(4.72)
Proof. For simplicity, the result is only shown for ω0 in the unconstrained
case—the remaining part for ω 0 is immediate. One starts from the definitions
of best possible estimates in the form
J(x0 )+ J(u + v) − J(u) v ω0 := sup
,
v2
0 +
J (x ) J(u + s · v) − J(u) v ω0 := sup
.
sv2
The extremal property of the Moore-Penrose pseudoinverse implies that
0 +
J (x ) J(u + v) − J(u) v ≤ J(x0 )+ J(u + v) − J(u) v ,
which directly implies ω0 ≤ ω0 .
Next, since
0 0 Δ (x , x ) ≤ Δ(x0 , x0 ) ,
(4.73)
an immediate consequence of the preceding lemma is that the associated
Kantorovich quantities
h0 := ω0 Δ (x0 , x0 ) , h0 := ω0 Δ(x0 , x0 )
satisfy the comparison property
h0 ≤ h0 .
Hence, by deliberate rank reduction, larger damping factors should be possible. In order to proceed to a theoretically backed choice of the damping
factors, computational estimates of the associated rank-deficient Lipschitz
constants are needed. Upon carefully revisiting the above full rank analysis
we may end up with the estimates:
4.3 Error Oriented Algorithms
217
1
Δ(xk−1 , xk ) − Δ(xk , xk ) + Δ(xk , xk−1 )2 − Δ(xk−1 , xk )2 2
[ω k ] :=
λk−1 Δ(xk−1 , xk−1 )Δ(xk−1 , xk )
(4.74)
wherein, for the unconstrained case, the quantities
Δ(xk , xk−1 ) := −J(xk )+ F (xk ) + J(xk−1 )Δ(xk−1 , xk ) ,
Δ(xk−1 , xk ) := P ⊥ (xk , xk )Δ(xk−1 , xk )
can be cheaply computed. Moreover, with either [δ] := 0 or
J(xk )+ r(xk−1 )
[δ] :=
< 1,
J(xk )+ F (xk−1 )
the trust region strategy for the rank-deficient case is complete: both the prediction strategy (4.71) and the correction strategy (4.69) remain unchanged—
with the proper identification of terms such as [ω k ] by (4.74) in the prediction
case.
An interesting question is, whether the above Lipschitz constant estimates
also inherit property (4.72).
Lemma 4.18 Let [ωk ], [ωk ] and [ω k ], [ω k ] denote the computational estimates as defined above, associated with the Jacobian matrices J(xk ) and
J (xk ), respectively. Then these quantities satisfy
[ωk ] ≤ [ωk ] , [ω k ] ≤ [ω k ] ,
where equality only holds, if the residual component dropped in the rank reduction process vanishes.
Proof. First, one observes that, in the unconstrained case
[ωk ] = J(xk )+ J(xk ) − J(xk−1 ) Δ(xk−1 , xk )
d
with d a short hand notation for the denominator in (4.70). After rank reduction, the corresponding expression is
[ω k ] = J (xk )+ J (xk ) − J(xk−1 ) Δ(xk−1 , xk ) d .
By means of the usual extremal property of the Moore-Penrose pseudoinverse,
one would immediately have that
k +
J (x ) J(xk ) − J(xk−1 ) Δ(xk−1 , xk )
≤ J(xk )+ J(xk ) − J(xk−1 ) Δ(xk−1 , xk ) .
Hence, in order to connect the numerators in [ω k ] and [ω k ], one needs to show
that the vectors
218
4 Least Squares Problems: Gauss-Newton Methods
z := J + Jy , z := J + J y
are identical for arbitrary y. For this purpose, consider the QR-decompositions
⎡
⎤
⎡
⎤
R11 R12
R11 R12
R22 ⎦ , J = QT ⎣ 0
0 ⎦.
J = QT ⎣ 0
0
0
0
0
Note that, as long as only the Euclidean norm arises, QR-decomposition
formally includes SV D-decomposition (they just differ by orthogonal transformations). Therefore the above representation models our above rank reduction strategy. Since the difference of Q and Q only acts on R22 , one may
also write:
⎡
⎤
R11 R12
0 ⎦.
J = QT ⎣ 0
0
0
Hence,
z
=
.
=
z1
z2
R11
0
.
=
R12
0
⎡
⎤
/+
R11 R12 R11 R12
x1
T ⎣
⎦
0
R22
QQ
0
0
x2
0
0
/+ .
/+ R11 x1 + R12 x2
R11 R12
u
=:
.
R22 x2
0
0
v
The associated linear least squares problem is
R11 z1 + R12 z2 − u2 + v2 = min
subject to
z = min .
This must be compared with the linear least squares problem defining z :
R11 z1 + R12 z2 − u2 = min
subject to
From this, one concludes that
z = min .
z = z .
One is now ready to further treat the numerator in [ω k ] by virtue of
k + k
J (x ) J (x ) − J(xk−1 ) Δ(xk−1 , xk )
= J (xk )+ J(xk ) − J(xk−1 ) Δ(xk−1 , xk ) .
Upon collecting all results, we finally have proven that
[ω k ] ≤ [ω k ] ,
4.3 Error Oriented Algorithms
219
where equality only holds, if the dropped residual component of the vector
J(xk ) − J(xk−1 ) Δ(xk−1 , xk )
actually vanishes. The same kind of argument also verifies that
[ωk ] ≤ [ωk ] ,
which completes the proof of the lemma.
In view of (4.72) and (4.73) together with the (technical) assumption δ := δ,
the estimated damping factors
1−δ
λk := min 1 ,
[ωk ]Δ (xk , xk )
should increase
λk ≥ λk ,
a phenomenon that has been observed in actual computation.
Summarizing, the above theoretical considerations nicely back the observations made in practical applications of such a rank strategy. Numerical experience, however, indicates that only rather restricted use of this device should
be made in realistic examples, since otherwise the iteration tends to converge
to critical points with a rank-deficient Jacobian or to trivial solutions that
are often undesirable in scientific applications. For this reason, a deliberate
rank reduction is recommended only, if
λk < λmin 1
for some prescribed default parameter λmin (say λmin = 0.01). In such a
situation, deliberate rank reduction may often lead to damping factors
λk ≥ λmin
at some intermediate iteration step and allow one to continue the iteration
with greater rank afterwards.
A convergence analysis comparable to the preceding sections would be extremely technical and less satisfactory than in the simpler cases treated before. As it turns out, the whole rank reduction device is rarely activated,
since the refined damping strategies anyway tend to avoid critical points with
rank-deficient Jacobian. The device does help, however, in inverse problems
whenever the identification of some unknown model parameters is impossible
from the comparison between assumed model and given data. In these cases,
at least local convergence is guaranteed by Theorem 4.7.
220
4 Least Squares Problems: Gauss-Newton Methods
Bibliographical Note. The possible enlargement of the convergence domain by means of deliberate rank reduction has been pointed out by the
author in 1972 in his dissertation [59, 60]. In 1980, the effect was partially
redetected by J. Blue [28].
The described adaptive trust region and rank strategies have been implemented in the codes NLSQ-ERR for unconstrained and NLSCON for equality
constrained nonlinear least squares problems.
Fig. 4.3. Parameter fit: X-ray spectrum. The background signal (primary
beam) of unknown shape gives rise to a rank-deficiency 2. Top: measurements,
center: fit to measurements, bottom: fit after subtraction of primary beam.
Example 4.2 X-ray spectrum. In order to illustrate the convergence of the
rank-deficient Gauss-Newton method, a rather typical example from X-ray
spectroscopy is given now. In this example, measurements of small angle
diffraction on the crystalline regions of collagen fibers are compared with a
4.4 Underdetermined Systems of Equations
221
model function representing a finite series of Gaussian peaks. The peaks of
actual interest are contaminated by a background signal (‘primary beam’) of
unknown shape.
Assessing a combination of Lorentz peaks, Gaussian peaks and a constant to
the background signal, Jacobian rank-deficiency naturally arises. The data
{yi } were obtained from an electron multiplier with a noise level characterized
by absolute measurement tolerances
δyi =
√
yi , i = 1, . . . , m .
Without explicit setting of these error tolerances, this problem had turned
out to be not solvable.
In Figure 4.3, the fit of n = 23 model parameters to given m = 178 measurement values is represented. In a neighborhood of the solution, a constant rank
deficiency of 2 occurred. Nevertheless, the Gauss-Newton method succeeded
in supplying a satisfactory solution.
It may be worth mentioning that standard filtering techniques, which just
separate high and low frequencies, are bound to fail in this example, since
the unwanted background signal contains both low and high frequencies.
4.4 Underdetermined Systems of Equations
In this section we discuss the iterative solution of underdetermined systems
of nonlinear equations
F (x) = 0 ,
(4.75)
where F : D ⊂ Rn → Rm with m < n. First, in Section 4.4.1, error oriented
local Gauss-Newton algorithms are studied in close connection with the treatment of overdetermined systems in Section 4.3. In particular, a local quasiGauss-Newton method is worked out including an appropriate extension of
Broyden’s ‘good’ update technique—compare Section 2.1.4 above. Next, in
Section 4.4.2, a globalization of these methods is derived.
4.4.1 Local quasi-Gauss-Newton method
This section deals with local Gauss-Newton methods, which will require ‘sufficiently good’ starting points.
Linear underdetermined systems. We start with a discussion of the
linear special case
Ax = b ,
222
4 Least Squares Problems: Gauss-Newton Methods
where A is an (m, n)-matrix and x, b are vectors of corresponding length. As
for the solution structure, we may adapt Lemma 4.1 of Section 4.1.1 to the
present situation: For p := rank(A) ≤ m < n, there exists some (n − p)dimensional solution subspace X ∗ . The general solution may be decomposed
as
x = x∗ + z
with arbitrary z ∈ ker(A) and x∗ the ‘shortest’ solution in X ∗ defined by
x∗ = A+ b
(4.76)
in terms of the Moore-Penrose pseudo-inverse of A. As in Section 4.1.1 we
define the orthogonal projectors P = A+ A, P = AA+ . In particular, we have
rank(A) = m < n ⇐⇒ P = Im , P
⊥
= 0.
As in the overdetermined case, the actual computation of (4.76) is done via
a QR-decomposition with column pivoting
QAΠ = [R, S] ,
(4.77)
where R is a nonsingular upper triangular (m, m)-matrix with diagonal entries ordered according to modulus as in (4.9) and S an (m, n − m)-matrix.
If m = n + 1, then the permutation Π naturally defines some external parameter, say ξ, as the variable corresponding to the last column S.
Let
v1
x = A+ b = Π T
v2
with partitioning v1 ∈ Rm , v2 ∈ Rn−m . For n − m m, which is the usual
case treated here, the QR-Cholesky decomposition [65] is recommended:
Backsubstitutions:
Rw = S ,
M := In−m + wT w ,
Inner products:
Cholesky decomposition:
Projection:
M = LLT ,
Rv1 = b ,
v 2 := wT v 1 ,
M v2 = v 2 ,
v1 = v1 − wT v2 .
The total computational amount is n−m square root evaluations and roughly
1 2
2 m (n − m) multiplications. If the row rank p must be reduced successively
within some adaptive rank strategy (compare Section 4.3.5), then some additional ∼ m(n − m) array storage places are needed.
Local Gauss-Newton method. We are now ready to treat the truly nonlinear case. Linearization of (4.75) leads to the underdetermined system
F (x)Δx = −F (x) .
4.4 Underdetermined Systems of Equations
223
From this we arrive at the local Gauss-Newton method (k = 0, 1, . . .):
Δxk = −F (xk )+ F (xk ) , xk+1 := xk + Δxk .
(4.78)
Again we define the orthogonal projectors P and P (assuming full row rank)
via
P (x) := F (x)F (x)+ = Im ,
P (x) := F (x)+ F (x) = In .
(4.79)
As for local convergence, a specialization of Theorem 4.7 will apply. Since
⊥
P (x) = 0 from (4.79), we may directly conclude that κ(x) ≡ 0 and therefore
drop assumption (4.38). Moreover, uniqueness of the solution now longer
holds. The associated convergence theorem for the underdetermined case then
reads:
Theorem 4.19 Let F : D ⊆ Rn → Rm , m < n, with D open, convex denote
a continuously-differentiable mapping. Consider the Gauss-Newton method
(4.78). Assume that one can find a starting point x0 ∈ D, and constants α,
ω ≥ 0 such that
F (x0 )+ F (x0 ) ≤ α ,
F (x)+ F (y) − F (x) (y − x) ≤ ωy − x2
for all x, y ∈ D collinear, y − x ∈ R F (x)+ .
Moreover, let
h := αω < 2 ,
S(x0 , ρ) ⊂ D with ρ := α/(1 − 12 h) .
Then:
(I) The sequence {xk } of Gauss-Newton iterates is well-defined, remains in
S(x0 , ρ) and converges to some x∗ ∈ S(x0 , ρ) with
F (x∗ )+ F (x∗ ) = 0 .
(II) Quadratic convergence can be estimated according to
xk+1 − xk ≤ 12 ωxk − xk−1 2 .
In passing, we want to emphasize that the local quadratic convergence property for the underdetermined case also implies that Jacobian approximation
errors up to considerable size are self-corrected—just as in the ordinary Newton method.
224
4 Least Squares Problems: Gauss-Newton Methods
Local quasi-Gauss-Newton method. For the underdetermined case a
variant of the error oriented Jacobian rank-1 update technique, the so-called
‘good’ Broyden method can be developed (cf. Section 2.1.4). In analogy to
formula (2.26), let J0 := F (x0 ) and define the update formula (k = 0, 1, . . .)
Jk+1 := Jk + F (xk+1 )
(δxk )T
δxk 22
(4.80)
together with the iteration
δxk := −Jk+ F (xk ) , xk+1 := xk + δxk .
(4.81)
This quasi-Gauss-Newton method has a nice geometric interpretation to be
derived now.
Lemma 4.20 Let N (Jk ) denote the nullspace of the Jacobian updates Jk ,
k = 0, 1, . . . with J0 := F (x0 ). Then the quasi-Gauss-Newton method (4.81)
is equivalent to the good Broyden method within the m-dimensional hyperplane
H := x0 ⊕ N ⊥ F (x0 ) = x ∈ Rn |x = x0 + P (x0 , x0 )z , z ∈ Rn .
Proof. From the definition (4.81), we may directly verify that
δxk ⊥ N ⊥ (Jk ) ,
which with the recursion (4.80) implies that
N (Jk+1 ) = N (Jk ) .
Hence, for k > 0
xk − x0 ∈ N ⊥ (J0 ) ,
which confirms the basic statement of the lemma.
The situation depicted in Figure 4.4 inspires an orthogonal coordinate transformation including some shift such that x0 is the new origin and H spans
Rm . In this coordinate frame, the quasi-Gauss-Newton correction δxk may
be interpreted as the quasi-Newton correction in H. At the same time, the
update formula (4.80) in Rn degenerates to the good Broyden update (2.26)
in Rm . This completes the proof of the Lemma.
From the above lemma we may directly conclude that, under reasonable assumptions (cf. Theorem 2.9), the quasi-Gauss-Newton iteration converges
superlinearly to some solution point x∗ , which is the intersection of the hyperplane H with the solution manifold—see also Figure 4.4. Moreover, the
quasi-Gauss-Newton iteration can be cheaply computed by a variant of algorithm QNERR of Section 2.1.4: just replace each solution of the linear system
4.4 Underdetermined Systems of Equations
225
x∗
N
M
H
x0
Fig. 4.4. Quasi-Gauss-Newton algorithm: convergence toward some x∗ at
intersection of solution manifold M and hyperplane H orthogonal to Jacobian
nullspace N .
J0 v = −F
by the solution of the underdetermined linear system
v = −J0+ F ,
which can be realized via the QR-Cholesky-decomposition.
This algorithm requires only the usual array storage for the decomposition
of J0 and, in addition, (kmax + 1)n extra storage places for δx0 , . . . , δxk .
Convergence monitor. We need a convergence criterion to assure ‘sufficiently good’ starting points. On the basis of Lemma 4.20 we can directly
adopt the criteria from the simplified Newton and the quasi-Newton method.
On this theoretical basis, we merely compute the usual contraction factors
Θk in terms of the quasi-Newton corrections. The iteration is terminated
whenever
δxk+1 2
Θk =
> 12 .
δxk 2
4.4.2 Global Gauss-Newton method
This section again deals with global Gauss-Newton methods, which (under
the usual mild assumptions) have no principal restriction on the starting
points. We proceed as in the case of the global Newton method and try first
to construct some Gauss-Newton path from an affine covariant level concept
and second, on this geometrical basis, an adaptive trust region strategy to be
realized via some damping of the Gauss-Newton corrections.
Geodetic Gauss-Newton path. The underdetermined nonlinear system
F (x) = 0 generically describes some (n − m)-dimensional solution manifold
M∗ in Rn . As in the case m = n, we argue that this systems is equivalent to
any system of the kind
226
4 Least Squares Problems: Gauss-Newton Methods
AF (x) = 0,
A ∈ GL(m) ,
which means that affine covariance is again a natural requirement for both
theory and algorithms. Therefore, as in Section 3.1.4, definition (3.20), we
introduce level sets G(x0 |A) at the initial guess x0 . Intersection of these level
sets according to
G(x0 ) :=
G(x0 |A)
A∈GL(m)
generates the λ-family of underdetermined mappings
Φ(x, λ) := F (x) − (1 − λ)F (x0 ) = 0 , λ ∈ [0, 1] ,
which generically will define a family of manifolds M(λ), λ ∈ [0, 1], such that
M∗ = M(1). Let y(λ) denote any path out of a continuum Y of paths, all of
which start at x0 and satisfy
Φ y(λ), λ ≡ 0 , λ ∈ [0, 1] .
Differentiation with respect to λ then yields an underdetermined linear system for the direction fields
F (y)ẏ = −F (x0 ) ,
which is equivalent to
P (y)ẏ = Δ(y, x0 ) ,
(4.82)
wherein P denotes the already introduced orthogonal projector and Δ is
defined as
Δ(y, x) := −F (y)+ F (x) .
In order to construct a locally unique path x(λ), we will naturally require the
additional local orthogonality condition
ẋ(λ) ⊥ N F x(λ) ,
which is equivalent to
P ⊥ (x)ẋ = 0 .
(4.83)
Upon adding (4.82) and (4.83), we end up with the unique representation
ẋ = Δ(x, x0 ) , x(0) = x0 .
of some path x. Its tangent direction in the starting point x0 is seen to be
ẋ(0) = Δ(x0 , x0 ) = −F (x0 )+ F (x0 ) ,
i.e., just the ordinary Gauss-Newton correction. A geometric picture of the
situation is given in Figure 4.5.
As can be verified without much technicalities, the path x(λ) is defined
uniquely: it starts at the given point x0 = x(0) and continues up to some
solution point x∗ = x(1) ∈ M∗ . Among all possible paths y ∈ Y the path x
exhibits an intriguing geodetic property to be shown in the following lemma.
4.4 Underdetermined Systems of Equations
227
x∗
x0
Fig. 4.5. Manifold family M(λ) and geodetic Gauss-Newton path x.
Lemma 4.21 Let Y denote the continuum of all C 1 -paths y satisfying
F y(λ) − (1 − λ)F (x0 ) = 0 , F y(λ) ẏ = −F (x0 ) .
Then the Gauss-Newton path x ∈ Y satisfies the minimal property
1
1
ẋ(λ)22 dλ
λ=0
= min
y∈Y
ẏ(λ)22 dλ .
λ=0
Proof. The minimal property can be shown to hold starting from the pointwise relation
ẏ(λ)22
= P (y)ẏ(λ)22 + P ⊥ (y)ẏ(λ)22
= Δ(y, x0 )22 + P ⊥ (y)ẏ(λ)22 ≥ ẋ(λ)22 .
Moreover, since x ∈ Y and P ⊥ (x)ẋ = 0 determines x uniquely, the above
result is essentially verified.
Because of the above geodetic minimal property, the path x will be called
geodetic Gauss-Newton path herein.
Adaptive trust region strategy. On the basis of the above geometrical
insight, a global Gauss-Newton method with affine covariant damping strategy
Δ(xk , xk ) = −F (xk )+ F (xk ) , xk+1 = xk + λk Δ(xk , xk )
can be constructed. An extended natural monotonicity test
Δ(xk , xk+1 )2 ≤ Δ(xk , xk )2 .
228
4 Least Squares Problems: Gauss-Newton Methods
is obtained to test for suitable damping factors λk . Within formula (4.74),
which is the basis for the prediction strategy (4.71) in the rank deficient case
(including the underdetermined case here), the term Δ(xk , xk−1 ) vanishes.
However, the replacement
Δ(xk−1 , xk ) − Δ(xk , xk ) −→ P (xk ) Δ(xk−1 , xk ) − Δ(xk , xk )
must be performed. Upon observing that P ⊥ (xk )Δ(xk , xk ) = 0, we end up
with the computational estimate [ω k ] ≤ ω k of the local Lipschitz constant ω k
as
1/2
Δ(xk−1 , xk ) − Δ(xk , xk )2 − P ⊥ (xk )Δ(xk−1 , xk )2
[ω k ] :=
λk−1 Δ(xk−1 , xk−1 )Δ(xk−1 , xk )
and, consequently, with the first trial value for the damping factor as
λ0k := min 1, 1/([ω k ] Δ(xk , xk )) .
The correction strategy for possibly refined trials λik , i = 1, 2, . . . remains
unchanged as in (4.69) based on the estimate (4.68).
The thus defined adaptive damping strategies work surprisingly well in practice. Unfortunately, as in the simpler case of nonlinear equations (m = n),
an associated global convergence theorem cannot be proved here: each new
Gauss-Newton iterate xk induces a new geodetic Gauss-Newton path with
a corresponding solution point x∗ ∈ M∗ —a structure that is prohibitive to
proving convergence. Fortunately, local quadratic convergence is guaranteed
by the above Theorem 4.19.
Exercises
Exercise 4.1 We consider the iterative behavior of the unconstrained nonlinear least squares functional
f (x) = F (x)22
during the Gauss–Newton iteration.
a) For the local Gauss–Newton iteration prove that
⊥
2
f (xk+1 ) ≤ P (xk )F (xk ) + 12 hk P (xk )F (xk )
in the notation of Section 4.2.1.
b) For the global Gauss–Newton iteration (notation of Section 4.2.2) prove
that
⊥
2
f (xk+1 ) ≤ P (xk )F (xk ) + 1 − 12 λ(1 + ρ) P (xk )F (xk ) ,
if the optimal damping factor λ = λk from (4.25) is used.
Exercises
229
Exercise 4.2 Revisit assumption (4.14) for the local convergence of GaussNewton methods in the residual framework. In particular, consider the small
residual factor ρ(x) in the limiting case x → x∗ .
a) Interpret this term as local curvature of the C 2 -manifold F (x) = F (x∗ ).
b) Compare the condition ρ(x∗ ) < 1 with the stability condition for nonlinear least squares problems as imposed by P.-Å. Wedin (see [196, 144]).
Exercise 4.3 In Newton’s method, sufficiently small Jacobian perturbations are known to still keep superlinear convergence—this is the so-called
self-correction property. Consider the ordinary Gauss-Newton method for
compatible nonlinear least squares problems, which is known to converge superlinearly: here Jacobian perturbations, even if they are ‘sufficiently small’,
deteriorate the iterative behavior to linear convergence. Analyze this convergence pattern in terms of theory.
Exercise 4.4
For unconstrained nonlinear least squares problems
F (x)2 = min ,
we consider a modification of Broyden’s rank-1 update technique. Starting
from some given J0 = J(x0 ), Jacobian updates are obtained by
Jk+1 := Jk − Jk Δxk+1
ΔxTk
,
Δxk 22
where xk+1 = xk + Δxk with
Δxk := −Jk+ F (xk ) , Δxk+1 := −Jk+ F (xk+1 ) .
a) Derive the kind of secant condition that is satisfied by this update.
b) Show that such a quasi-Gauss-Newton iteration solves the ‘wrong’ nonlinear least squares problem
P 0 F (x)2 = min ,
where P 0 = J0 J0+ is an orthogonal projector.
Exercise 4.5
corrections
Consider a general steepest descent method based on the
Δx(A) = − grad(T (x|A)
230
4 Least Squares Problems: Gauss-Newton Methods
in terms of the general level function
T (x|A) := 12 AF (x)22 .
Let the Jacobian J be nonsingular, say, of the form
.
/
R S
J=
, R nonsingular (p, p)-matrix .
0 0
Let v := R−1 S ∈ Rn−p , and define the projectors
Ip 0
P :=
, P ⊥ = In − P .
0 0
a) Show that, for all nonsingular matrices A, the corrections Δx(A) are
confined to the (n − p)-dimensional hyperplane
(Hp )
P ⊥ Δx − v T P Δx = 0 .
b) Show that the associated Levenberg-Marquardt correction (3.17) also lies
in Hp .
c) Appropriate projection of Newton’s method yields
(P N )
P Δx + vP ⊥ Δx = −R−1 P F (x) ,
(P ⊥ N )
0 = −P ⊥ F (x) .
In general, the relation (P ⊥ N ) is a contradiction. Verify, however, that
the intersection
(Hp ) ∩ (P N )
leads to the Gauss-Newton correction
Δx = −J + F (x) .
Hint: Compare the QR-Cholesky decomposition for the rank-deficient
Moore-Penrose pseudo-inverse.
Exercise 4.6 We again consider the (possibly shortest) solution of the
weighted linear least squares problem
Bx − d22 + μ2 Ax − c22 = min ,
defined by some penalty parameter μ. In Section 4.1.2 we had presented the
treatment using Householder transformations. Here we study the numerical
solution via orthogonalization using fast Givens rotations—for details see,
e.g., [107].
Perform the penalty limiting process μ → ∞ for this case. What kind of
elimination method arises for the equality constraints? What kind of pivoting
is required? Which two sub-condition numbers arise naturally? How can these
be interpreted? Treatment of rank-deficiency in either case?
Exercises
Let
Exercise 4.7
231
A− := LB + R
denote a generalized inverse, defined via the Moore-Penrose pseudoinverse
B + with two nonsingular matrices L, R. Derive a set of four axioms which
uniquely defines A− .
Hint: Start from the four Penrose axioms for B.
Exercise 4.8 Separable Gauss-Newton method. We compare the two variants suggested by Golub-Pereyra [105] and by Kaufman [128]. Let αGP , ωGP ,
κGP denote the best possible theoretical estimates in Theorem 4.7 when applied to the Golub-Pereyra suggestion (4.21) and αK , ωK , κK the corresponding quantities for the Kaufman variant (4.22).
Assuming best possible estimates, prove that
αK ≤ αGP ,
ωK ≤ ωGP ,
κK (v ∗ ) ≤ κGP (v∗ ) .
Exercise 4.9 Show that the results of Theorem 4.7 also hold, if assumption
(4.38) is replaced by
F (x + v)− − F (x)− F (x) ≤ κ1 (x)v, x ∈ D .
Note that in this case the projection property (4.37) is not required to hold.
Exercise 4.10
Consider underdetermined nonlinear systems
F (x) = 0 ,
F : D ⊂ Rn −→ Rm ,
n>m
with generically nonunique solution x∗ . Verify that the residual based affine
contravariant Newton-Mysovskikh theorem (Theorem 2.12) still holds, if only
the iterative corrections Δxk solve the underdetermined linearized systems
F (xk )Δxk = −F (xk ) .
Interpret this result in terms of the uniqueness of the residual F (x∗ ).
5 Parameter Dependent Systems:
Continuation Methods
In typical scientific and engineering problems not only a single isolated nonlinear system is to be solved, but a family of problems depending on one or
more parameters λ ∈ Rp , p ≥ 1. The subsequent presentation will be mainly
restricted to the case p = 1 (with the exception of Section 4.4). In fact,
parameter dependent systems of nonlinear equations
F (x, λ) = 0,
x ∈ D ⊆ Rn ,
λ ∈ [0, L]
(5.1)
are the basis for parameter studies in systems analysis and systems design,
but can also be deliberately exploited for the globalization of local Newton or
Gauss-Newton methods, if only poor initial guesses are available.
In order to understand the structure of this type of problem, assume that
(5.1) has a locally unique solution (x∗ , λ∗ ) ∈ D × [0, L]. Let the (n, n)-matrix
Fx (x, λ) be regular in some neighborhood of this point. Then, by the implicit
function theorem, there exists a unique homotopy path x defined by virtue of
the homotopy
F x(λ), λ ≡ 0 , λ ∈ [0, L]
or, equivalently, by the linearly implicit ODE, often called the Davidenko
differential equation (in memory of the early paper [48] by D. Davidenko),
Fx ẋ + Fλ = 0
(5.2)
with a selected solution x∗ on the homotopy path as initial value, say
x(λ∗ ) := x∗ .
Note that the ODE (5.2) uniquely defines the direction field ẋ in terms of
the λ-parametrization.
In order to avoid the specification of the parametrization, one may introduce
the augmented variable
y := (x, λ) ∈ Rn+1
and rewrite the above mapping (5.1) as
F (y) = 0
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance
and Adaptive Algorithms, Springer Series in Computational Mathematics 35,
DOI 10.1007/978-3-642-23899-4_5, © Springer-Verlag Berlin Heidelberg 2011
233
234
5 Parameter Dependent Systems: Continuation Methods
and the direction field (5.2) as
F (y) t(y) = 0 .
Whenever the condition
or, equivalently,
rank F (y) = n
dim ker F (y) = 1 ,
then t(y) is uniquely defined up to some normalization, which might be fixed
as
t2 = 1 .
In general, whenever the local condition
rank F (y) = n − k
or, equivalently
dim ker F (y) = k + 1
holds, then a singularity of order k occurs. A special role is played by turning
points, which, with respect to the selected parameter λ, can be characterized
by k = 0 and
rank F (y ∗ ) = n ,
rank Fx (x∗ , λ∗ ) = n − 1 ,
so that they are, formally speaking, singularities of order k = 0. For k > 0 the
local direction field is not unique, its actual structure depending on properties
of higher derivatives up to order k + 1. For k = 1 simple bifurcation points
may occur, which require, however, some second derivative discriminant D to
be positive: in this case, two distinct branch directions are defined; if D = 0,
a so-called isola occurs. For k > 1 there exists a hierarchy of critical points,
which we cannot treat here in full beauty. The complete solution structure
of parameter dependent mappings, usually represented within a bifurcation
diagram, may turn out to be rather complicated. Here we will restrict our
attention to turning points and simple bifurcation points.
Every now and then, the scientific literature contains the suggestion to just
integrate the Davidenko differential equation (5.1) numerically, which is not
recommended here for the following reasons:
• In most applications only approximations of the Jacobian (n, n)-matrix
Fx (x, λ) are available.
• The numerical integration of (5.1) requires some implicit or at least linearly implicit discretization, which, in turn, requires the solution of linear
equations of the kind
Fx (y) − βΔλFλx (y) Δx + βΔλFxx [ẋ, Δx] = −ΔλFλ (y) ,
5 Parameter Dependent Systems: Continuation Methods
235
obviously requiring second order derivative information. Even though any
stiff integrators will solve the ODE, they will not assure the basic condition
F = 0 up to sufficient accuracy due to global error propagation—this is
the well-known ‘drift’ of the global discretization error.
Rather so-called discrete continuation methods are the methods of choice:
they concentrate on the solution of F = 0 directly and only require sufficiently accurate evaluation of the mapping F and approximations of F .
Such methods consist of two essential parts:
• a prediction method that, from given solution points (xν , λν ) on the hoν+1 ) assumed to be ‘sufmotopy path, produces some ‘new’ point (
xν+1 , λ
ficiently close’ to the homotopy path,
ν ),
• an iterative correction method that, from a given starting point (
xν , λ
supplies some solution point (xν , λν ) on the homotopy path.
For the prediction step, classical or tangent continuation are the canonical
choices—see below. Needless to say that, for the iterative correction steps, we
here concentrate on local Newton and quasi-Newton methods (see Sections
2.1.1 and 2.1.4 above) as well as (rank-deficient) Gauss-Newton methods (see
Section 4.4.1 above).
Bibliographical Note. The principle of local continuation has been suggested in 1892 by H. Poincaré [168] in the context of analytical continuation.
The idea of discrete continuation seems to date back to E. Lahaye [142] in
1934. As for the analysis of higher order singular points, the interested reader
may want to look up, e.g., the textbooks [109, 110, 111] of M. Golubitsky and
coauthors.
Since the underlying homotopy path is a mathematical object in the domain
space of the nonlinear mapping F , we select the affine covariant framework.
In Section 5.1 below, we derive an adaptive pathfollowing algorithm as a
Newton continuation method, which terminates locally in the presence of
critical points including turning points. In the next Section 5.2, based on the
preceding Section 4.4, we treat an adaptive quasi-Gauss-Newton continuation
method. This method is able to follow the path beyond turning points, but
still terminates in the neighborhood of any other critical point. In order to
overcome such points as well, we exemplify a scheme to construct augmented
systems, whose solutions are just selected critical points of higher order—
see Section 5.3. This scheme is an appropriate combination of LyapunovSchmidt reduction and topological universal unfolding. Details of numerical
realization are only worked out for the computation of diagrams including
simple bifurcation points.
Before we begin with a presentation of any algorithmic details, we want to
point out that, quite often, there is a choice of embedding to be made in view
of computational complexity.
236
5 Parameter Dependent Systems: Continuation Methods
Example 5.1
Choice of embedding. Consider the problem from [99]
G(x) := x − φ(x) = 0,
with
10
φi (x) := exp(cos(i ·
xj )),
i = 1, . . . , 10.
j=1
In [99], K. Georg treated the unspecific embedding
F (x, λ) = λF (x) + (1 − λ)x = x − λφ(x) .
(5.3)
Assume that we know the solution at λ = 0, which is x0i = 0, i = 1, . . . , 10,
and want to find the solution at λ = 1. The solution structure x(λ) is given
in Figure 5.1, left.
Alternatively, we might choose the more problem-oriented embedding (compare also [77, Section 4.4])
10
F̃i (x, λ) := xi − exp(λ · cos(i ·
xj )),
i = 1, . . . , 10 .
(5.4)
j=1
The solution at λ = 0 is now given by x0i = 1, i = 1, . . . , 10. The corresponding
solution structure as given in Figure 5.1, right, is obviously much simpler.
2.5
2.0
1.8
2.0
1.6
1.4
1.5
1.2
1.0
1.0
0.8
0.5
0.6
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 5.1. Example 5.1. Left: unspecific embedding (5.3). Right: problem-oriented
embedding (5.4).
All computations have been performed by the Gauss-Newton continuation
code ALCON1 to be described in Section 5.2 below. The dots in Figure 5.1
indicate the number of discrete continuation points as obtained from ALCON1:
Observe that the computational complexity on the left is much higher than
on the right. Look also at the quite different number of turning points. The
cross-points arise from the projection x9 (λ), not from bifurcations.
5.1 Newton Continuation Methods
237
5.1 Newton Continuation Methods
This section deals with the situation that there exists a unique homotopy path
x that can be explicitly parametrized with respect to λ over a finite interval
of interest. A confirmation of this structure may often come directly from
expert insight into the scientific or engineering problem to be solved. In this
case the Jacobian (n, n)-matrix Fx (x, λ) is known to be nonsingular, which
excludes the occurrence of any type of critical points. As a consequence, local
Newton algorithms may well serve as iterative correction methods within any
discrete continuation method.
In order to treat the problem family (5.1), a sequence of problems
F (x, λν ) = 0 , ν = 0, 1 . . . ,
(5.5)
is solved instead, where the interval [0.L] is replaced by the subdivision
0 = λ0 < λ1 < · · · < λN = L .
In order to solve each of the problems (5.5) by a local Newton method,
‘sufficiently good’ starting points are required, which should be supplied by
some suitable prediction method. Formally speaking, any starting points will
lie on some prediction path x
(λ) for λ = λν . The task therefore involves the
choice of the prediction method (Section 5.1.1), the theoretical analysis of
the coupling between prediction and Newton method (Section 5.1.2), which
leads to a characterization of feasible stepsizes, and, on this theoretical basis,
the adaptive choice of the stepsizes Δλν = λν+1 − λν in actual computation
(Section 5.1.3). Since paths as mathematical objects live in the domain space
of the mapping F , the affine covariant setting for both theory and algorithms
is selected throughout Section 5.1.
5.1.1 Classification of continuation methods
As the first idea to choose a suitable starting point x
(λν+1 ) one will just take
the previous solution point x(λν ). This so-called classical continuation method
is represented schematically in Figure 5.2. For this continuation method the
prediction path is defined as
x
(λ) = x(λν ) , λ ≥ λν .
A refinement of the above idea is to proceed along the tangent of the homotopy path in λν . This is the so-called tangent continuation method, sometimes
also called method of incremental load or Euler continuation, since it realizes
the explicit Euler discretization of the ODE (5.2). The corresponding scheme
is depicted in Figure 5.3. The associated prediction path is defined by
x
(λ) = x(λν ) + (λ − λν ) ẋ(λν ) , λ ≥ λν ,
238
5 Parameter Dependent Systems: Continuation Methods
wherein
−1 ẋ(λν ) = −Fx x(λν ), λν
Fλ x(λν ), λν .
Note that both the classical and the tangent prediction paths are given in
affine covariant terms, which suggests that a theoretical classification of prediction methods should also be formulated in such terms that match with the
error oriented local convergence analysis of Newton’s method from Section
2.1 above.
Definition. Let Δλ := λ − λν . A continuation method defined via the prediction path x
(λ) is said to be of order p, if a constant ηp exists such that
x(λ) − x
(λ) ≤ ηp · Δλp .
(5.6)
In order to illustrate this definition, a few examples are given first. For simplicity, let λν := 0 and λ = Δλ.
Classical continuation method. For the method represented in Figure 5.2
one immediately derives
x(λ) − x
(λ) = x(λ) − x(0) ≤ λ · max ẋ(s) .
s∈[0,L]
Hence, this method is of the order p = 1 with order coefficient
η1 := max ẋ(s) .
s∈[0,L]
Actually, both H. Poincaré [168] and E. Lahaye [142] had just thought of this
simplest type of continuation.
x
x(λν+1 )
x
x(λν )
x̂(λν+1 )
x̂
λ
λν
λν+1
Fig. 5.2. Classical continuation method.
5.1 Newton Continuation Methods
239
Tangent continuation method. For the method represented in Figure 5.3
one obtains
x(λ) − x
(λ) = x(λ) − x(0) − λẋ(0) ≤ 12 λ2 max ẍ(s) .
(5.7)
s∈[0,L]
So, this method is of the order p = 2 with order coefficient
η2 := 12 max ẍ(s) .
(5.8)
s∈[0,L]
x̂(λν+1 )
x
x̂
x
x(λν+1 )
x(λν )
λ
λν
λν+1
Fig. 5.3. Tangent continuation method.
Standard embedding. The simple embedding
F0 (x, λ) := F (x) − (1 − λ)F (x0 )
is rather popular. Note, however, that this homotopy ‘freezes’ the information
of the starting point x0 . The least improvement, which can easily be realized,
is to turn to a damping or trust region strategy (see Chapter 3), which may
be understood as being formally based on the homotopy chain (k = 0, 1, . . .)
Fk (x, λ) := F (x) − (1 − λ)F (xk ) ,
which brings in the information about the ‘newest’ iterate xk . Observe, however, that this generally applicable homotopy does not exploit any special
structure of the mapping.
Partial standard embedding. In the experience of the author the only
sometimes successful variant has been the selection of only one component
of the mapping, which leads to the so-called partial standard embedding
F 0 (x, λ) := P F (x) + P ⊥ F (x) − (1 − λ)F (x0 ) ,
(5.9)
240
5 Parameter Dependent Systems: Continuation Methods
where P is an orthogonal projector and P ⊥ its complement. In what follows we give some comparative results on the classical versus the tangent
continuation method for this special kind of embedding.
Lemma 5.1 Consider the partial standard embedding (5.9). Notations as
introduced in this section. Then, with the Lipschitz condition
−1 F x
(λ)
F (x) − F (
x(λ)) 0 x − x
(λ) ,
≤ω
x, x
(λ) ∈ D , 0 ≤ λ ≤ L ,
the order coefficient for the classical continuation method is
−1
⊥
0 η1 = sup F x(λ)
P F (x ) ,
(5.10)
λ∈[0,λ̄]
the one for the tangent continuation is closely related as
η2 := 12 ω
0 η12 .
(5.11)
Proof. The extremely simple form of the embedding (5.9) leads to
∂
∂
∂
F 0 (x, λ) =
F (x) = F (x) ,
F 0 (x, λ) = P ⊥ F (x0 ) ,
∂x
∂x
∂λ
which implies
ẋ(λ) = −F (x)−1 P ⊥ F (x0 )
and therefore (5.10). For the estimation of η2 , we must start with a Lipschitz
condition for the directions
−1
−1
⊥
0 ẋ(λ) − ẋ(0) = F (x(λ)
−
F
x(0)
)P
F
(x
)
−1 ẋ(λ)
≤
F
x
(0)
F
(x(λ))
−
F
(x(0))
≤ω
0 x(λ) − x(0) η1 ≤ ω
0 η12 λ .
Hence, with (5.8), one has the second result (5.11), which completes the proof.
As for the consequence of this Lemma for the feasible stepsizes see Lemma
5.6 below.
It is an easy exercise to construct further refinements of prediction methods
beyond the tangent continuation method, just based on higher derivative
information of F . In view of complex real life problems, however, this is not
very promising, since this would also require accurate higher order derivative
information, which may be rarely available.
5.1 Newton Continuation Methods
241
Polynomial continuation In order to classify such methods, the monomials
Δλp in (5.6) must be replaced by some strictly monotone increasing function
ϕ(Δλ) with ϕ(0) = 0 such that
x(λ) − x
(λ) ≤ η · ϕ(Δλ) .
(5.12)
In order to illustrate this definition, we consider two extrapolation methods
(once more λν := 0 and λ := Δλ).
Standard polynomial extrapolation. Based on the data
x(λ−q ), . . . , x(λ−1 ) , x(0)
a prediction path can be defined as (with λ0 := 0)
0
x
q (λ) :=
x(λm )Lm
q (λ)
m=−q
in terms of Lagrange polynomials L(·). Application of standard approximation error estimates then leads to
x(λ) − x
(λ) ≤ Cq+1 · λ(λ − λ−1 ) · . . . · (λ − λ−q ) ,
which naturally defines
ϕ(λ) := λ(λ − λ−1 ) · · · · · (λ − λ−q ) .
(5.13)
The numerical evaluation of the prediction path is, of course, done by the
Aitken-Neville algorithm.
Hermite extrapolation. This type of polynomial extrapolation is based on the
data
x(λ−q ) , ẋ(λ−q ), . . . , x(0) , ẋ(0) .
Evaluation of the prediction path x
q (λ) is done here by a variant of the
Aitken-Neville algorithm for pairwise confluent nodes. Proceeding as above
leads to
x(λ) − x
(λ) ≤ C q+1 · λ2 (λ − λ−1 )2 · · · · · (λ − λ−q )2 ,
which defines the monotone function
ϕ(λ) := λ2 (λ − λ−1 )2 · · · · · (λ − λ−q )2 .
Note, however, that this prediction method requires quite accurate derivative
information, which restricts its applicability.
Bibliographical Note. An affine contravariant definition of the order of
a continuation method based on the residual F has been given in 1976 by
R. Menzel and H. Schwetlick [150]. The here presented affine covariant alternative is due to the author [61] from 1979. Its extension to polynomial
extrapolation is due to H.G. Bock [32]. For the standard embedding, convergence proofs were given by H.B. Keller [131] or M.W. Hirsch and S. Smale
[119]; a general code has been implemented by A.P. Morgan et al. [154].
242
5 Parameter Dependent Systems: Continuation Methods
5.1.2 Affine covariant feasible stepsizes
Any of the discrete continuation methods are efficient only for ‘sufficiently’
small local stepsizes
Δλν := λν+1 − λν ,
which must be chosen such that the local Newton method starting at the
predicted value x0 = x
(λν+1 ) can be expected to converge to the value x∗ =
x(λν+1 ) on the homotopy path. In what follows, a theoretical analysis of
feasible stepsizes is worked out, which will serve as the basis for an adaptive
stepsize control to be presented in Section 5.1.3.
Among the local Newton methods to be discussed as correction methods are
• the ordinary Newton method with a new Jacobian at each iterate (cf.
Section 2.1.1),
• the simplified Newton method with the initial Jacobian throughout (cf.
Section 2.1.2), and
• the quasi-Newton method starting with an initial Jacobian based on ‘good’
Broyden Jacobian updates (cf. Section 2.1.4).
Ordinary Newton method. We begin with the ordinary Newton method
as correction method within any discrete continuation method. The simplest
theoretical framework is certainly given by Theorem 2.3.
Theorem 5.2 Notation as introduced in this Section. Let Fx (x, λ) be nonsingular for all (x, λ) ∈ D × [0, L]. Let a unique homotopy path x(λ) exist
in a sufficiently large local domain. Assume the affine covariant Lipschitz
condition
Fx (x, λ)−1 (Fx (y, λ) − Fx (x, λ)) (y − x) ≤ ωy − x2 .
(5.14)
Let x
(λ) denote a prediction method of order p as defined in (5.6) based on
the previous solution point x(λν ). Then the ordinary Newton method with
starting point x
(λν+1 ) converges towards the solution point x(λν+1 ) for all
stepsizes
1/p
2
Δλν ≤ Δλmax :=
(5.15)
ωηp
within the interval [0, L].
Proof. Upon skipping any fine structure of the local domains assumed to
be sufficiently large, we must merely check the hypothesis of Theorem 2.3
for the ordinary Newton method with starting point x0 = x
(λ) and solution
point x∗ = x(λ). In view of the local contractivity condition (2.9), we will
estimate
5.1 Newton Continuation Methods
243
x∗ − x0 = x(λ) − x
(λ) ≤ ηp Δλp .
Upon inserting the upper bound into (2.9), we arrive at the sufficient condition
ηp Δλp < 2/ω ,
which is essentially (5.15).
The above theorem requires some knowledge about the ‘rather global’ Lipschitz constant ω. In order to permit a finer tuning within the automatic
stepsize control to be derived in Section 5.1.3, we next apply the affine covariant Newton-Kantorovich theorem (Theorem 2.1), which requires ‘more
local’ Lipschitz information.
Theorem 5.3 Notation and assumptions essentially as just introduced. Compared to the preceding theorem replace the Lipschitz condition (5.14) by the
condition
x(λ), λ)−1 Fx (y, λ) − Fx (x, λ) ≤ ω
0 y − x , x, y ∈ D .
(5.16)
Fx (
Then the ordinary Newton method with starting point x
(λν+1 ) converges towards the solution point x(λν+1 ) for all stepsizes
√
Δλν ≤ Δλmax :=
2−1
ω
0 ηp
1/p
.
(5.17)
Proof. The above Lipschitz condition (5.16) permits the application of Theorem 2.1, which (with λ = Δλ) requires that
α(λ)
ω0 ≤
1
2
.
(5.18)
So an upper bound Δx0 (λ) ≤ α(λ) for the first Newton correction needs
to be derived. To obtain this, we estimate
−1
Δx0 (λ) = Fx (
x(λ), λ) F (
x(λ), λ) = Fx (
x, λ)−1 (F (
x, λ) − F (x, λ))
= Fx (
x, λ)−1
1
Fx (x + s(
x − x), λ)(
x − x)ds s=0
≤ x − x 1 + 12 ω
0 x − x .
The application of the triangle inequality in the last step appears to be unavoidable.
With the definition of the order of a prediction method we are now able to
derive the upper bound
244
5 Parameter Dependent Systems: Continuation Methods
Δx0 (λ) ≤ ηp · λp 1 + 12 ω
0 ηp λp =: α(λ) .
Finally, combination of (5.18) and (5.19) yields
ω
0 ηp λp 1 + 12 ω
0 ηp λp ≤
or, equivalently
ω
0 ηp λp ≤
√
(5.19)
1
2
2−1,
which confirms the result (5.17).
Corollary 5.4 If the characterization (5.6) of a prediction method is replaced by the generalization (5.12) in terms of a strictly monotone increasing
function ϕ(Δλ), then the maximum feasible stepsize (5.17) is replaced by
√
2
−
1
Δλmax := ϕ−1
(5.20)
ω
0 η
with ϕ−1 the mapping inverse to ϕ.
Proof. Instead of (5.19), we now come up with (once again λ := Δλ)
α(λ) := ηϕ(λ) 1 + 12 ω
0 ηϕ(λ) ,
which directly leads to
ω
0 ηϕ(λ) ≤
√
2−1
thus confirming (5.20).
Simplified Newton method. In most applications, the simplified Newton
method rather than the ordinary Newton method will be realized. For this
specification, the following slight modifications hold.
Corollary 5.5 Notation and assumptions as in Theorem 5.3 or Corollary 5.4,
respectively. Let the ordinary Newton method therein be replaced by the simplified Newton method with the same starting point. Then, with the mere
replacement of the Lipschitz constant ω
0 via the slightly modified condition
−1
x(λ), λ)
Fx (x, λ) − Fx (
x(λ), λ) ≤ ω
0 x − x
(λ) ,
Fx (
the maximum feasible stepsizes (5.17) and (5.20) still hold.
Proof. The above two proofs can be essentially copied: the contraction condition (2.3) of Theorem 2.1 needs to be formally replaced by condition (2.17)
of Theorem 2.5, which means the mere replacement of the Lipschitz condition
(2.2) by (2.16).
5.1 Newton Continuation Methods
245
Partial standard embedding. In order to illustrate what has been said
at the end of Section 5.1.1 about the embedding (5.9), we now study the
consequences of Lemma 5.1 for the associated feasible stepsizes.
(1,2)
Lemma 5.6 Notation and assumptions as in Lemma 5.1. Let Δλmax denote
the maximum feasible stepsizes for the classical continuation method (p = 1)
and for the tangent continuation method (p = 2). Then the following results
hold:
√
(1)
Δλ(2)
=
2( 2 + 1)Δλ(1)
max
max ≈ 2.2 Δλmax .
Proof. We merely insert the results (5.10) for p = 1 and (5.11) for p = 2 into
the maximum stepsize formula (5.17) to verify the above results.
Obviously, in this rather unspecific embedding, tangent continuation seems
to require roughly double the number of continuation steps compared to classical continuation. This theoretically backed expectation has actually been
observed in large scale problems. At the same time, however, an efficient implementation of the tangent continuation method will roughly require double
the amount of work per step (see Section 5.1.3). Hence, there is no clear
advantage on either side. In sensitive examples, however, smaller stepsizes
increase robustness and reliability of the numerical pathfollowing procedure.
That is why for this type of embedding classical continuation is generally
recommended.
5.1.3 Adaptive pathfollowing algorithms
In the preceding section we analyzed discrete continuation methods with the
simplified Newton method as correction method. In its actual realization in
the code CONTI1, this is extended to some quasi-Newton correction method.
Simplified Newton method. This method keeps the Jacobian matrix
F (x0 ) fixed for all iterative steps, which implies that a single matrix decomposition at the beginning is sufficient throughout the iteration. As a
convergence monitor, the contraction factors Θk in terms of the simplified
Newton corrections are used. From (2.21) we require the local convergence
criterion
Θ0 ≤ Θ = 14 ,
which is easily tested after the first Newton step.
Quasi-Newton method. After the first iterative step, we substitute the
simplified Newton iteration by the quasi-Newton method based on ‘good’
Broyden updates, as documented by algorithm QNERR in Section 2.1.4. Let
Θk denote the corresponding contraction factors in terms of the quasi-Newton
246
5 Parameter Dependent Systems: Continuation Methods
corrections. In agreement with the convergence analysis in Theorem 2.7 we
require that Θk ≤ 1/2. Hence, whenever the condition
Θk >
1
2
(5.21)
occurs, then the continuation step λν → λν + Δλν is repeated with reduced
stepsize Δλν .
Discrete continuation method. The classical continuation method is certainly most simple to realize and needs no further elaboration. The tangent
continuation method additionally requires the numerical solution of a linear
system of the kind
Fx (x(λν ), λν )ẋ(λν ) = −Fλ (x(λν ), λν ) ,
which has the same structure as the systems arising in Newton’s method.
Note, however, that in order to realize a continuation method of actual order
p = 2, sufficiently accurate Jacobian approximations (Fx , Fλ ) need to be
evaluated not only at the starting points x
, but also at the solution points x.
In large scale problems, this requirement may roughly double the amount of
work per continuation step compared to the classical method (p = 1).
Adaptive stepsize control. In the globalization of local Newton methods
by continuation, adaptive trust region strategies come up as adaptive stepsize strategies. Colloquially, the construction principle is as follows: choose
stepsizes such that the initial guess x
(λν ) stays within the ‘Newton contraction tube’ around the homotopy path—see the theoretical stepsize results like
(5.17) containing the unavailable theoretical quantities ω
0 and ηp .
Following once more our paradigm of Section 1.2.3, we replace the unavailable
theoretical quantities by computationally available estimates [
ω0 ] ≤ ω
0 and
[ηp ] ≤ ηp —thus arriving at stepsize estimates
√
[Δλmax ] :=
2−1
[
ω0 ][ηp ]
1/p
≥ Δλmax .
(5.22)
Again both a prediction strategy and a correction strategy will be needed.
Of course, all formulas will be invariant under rescaling or shifting of the
continuation parameter.
Suppose that, for given λν+1 , the value Θ0 has already been computed. From
the convergence analysis of the simplified Newton method we know that
0
Θ0 (λ) ≤ 12 ω
0 Δx (λ) ≤ 12 ω
0 α(λ) .
Insertion of α(λ) from (5.19) yields
Θ0 ≤ 12 ω
0 ηp Δλp 1 + 12 ω
0 ηp Δλp
(5.23)
5.1 Newton Continuation Methods
247
or, equivalently,
ω
0 ηp Δλp ≥ g(Θ0 )
in terms of the monotone increasing function
√
g(Θ) := 1 + 4Θ − 1 .
From this, we may obtain the a-posteriori estimate
g Θ0 (λ)
[
ω0 ηp ] :=
≤ω
0 ηp
Δλp
and the associated stepsize estimate
1/p
g Θ
[Δλmax ] :=
.
[
ω0 ηp ]
√
Note that g Θ = 2 − 1 as in formula (5.17)—a mere consequence of the
fact that both formulas are based on the Kantorovich condition. Let Δλν
denote some desirable stepsize corresponding to Θ0 = Θ, whereas the actual
stepsize Δλν corresponds to the actually computed value of Θ0 . Then the
above derivation leads to the stepsize correction formula
Δλν
1/p
g Θ
:=
Δλν .
g(Θ0 )
(5.24)
For Θ0 < Θ the actual stepsize Δλν is acceptable, since Δλν > Δλν . If,
however, the termination criterion (5.21) is activated by some Θk > 1/2,
then the last continuation step should be repeated with stepsize
Δλν
1/p
g Θ
:=
Δλν ,
g(Θk )
which is a clear reduction, since
√
Δλν <
2−1
√
3−1
1/p
Δλν ≈ 0.571/p Δλν .
In order to derive a-priori estimates, we may exploit (5.23) again to obtain
[
ω0 ] :=
2Θ0 (λν )
0
Δx (λν )
≤ω
0
and just use the definition of the order of a prediction method in the form
248
5 Parameter Dependent Systems: Continuation Methods
x(λν ) − x(λν )
≤ ηp .
|Δλν−1 |p
[ηp ] :=
Upon inserting these quantities into (5.22), a stepsize prediction strategy is
defined via
1/p
0
g Θ
Δx (λν )
0
Δλν :=
·
Δλν−1 .
(5.25)
x(λν ) − x(λν ) 2Θ0
Note that this estimate is not sensitive to the computational accuracy of x.
Even if only a single Newton step is performed, which means that
0
x→x
:= x
(λν ) + Δx (λν ) ,
then the predicted value degenerates to
1/p
g Θ
Δλ0ν :=
Δλν−1 ,
2Θ0
which, compared with (5.24), is seen to be still a reasonable estimate. Precaution must be taken in the nearly linear case, characterized by
Θ0 ≤ Θmin 1 .
In this case, the stepsize estimate (5.25) should be replaced by
Δλ0ν
1/p
g Θ
:=
Δλν−1 .
2Θmin
to avoid exponential overflow.
Polynomial continuation. Both the correction and the prediction strategy
carry over to the more general case of continuation by polynomial extrapolation. Upon recalling definition (5.12), formula (5.24) must be modified such
that
g Θ
−1
Δλν = ϕ
ϕ(Δλν ) .
g (Θ0 )
A comparable formula holds instead of (5.25). The above required evaluation
of ϕ−1 is easily performed: since the typically arising functions ϕ are convex
(see, e.g., (5.13)), both the ordinary and the simplified Newton method in
R1 converge certainly monotonically; good starting guesses are available from
the dominant monomial so that the methods converge even fast.
5.1 Newton Continuation Methods
249
Graphical output. The automatically selected points typically give a really
good representation on the basis of comparably few data—which is a nice side
effect of any efficient stepsize control.
Classical continuation. In this case the data x(λν ) are available, so that in the
interactive mode only linear interpolation is possible, whereas in the batch
mode cubic spline interpolation will be preferable. For details the reader may
check, e.g., Section 7.4 in the textbook [77].
Tangent continuation. In this case both the nodal values x(λν ) and the associated tangents ẋ(λν ) are available data so that Hermite interpolation will
be the method of choice—compare, e.g., Section 7.1.2 in [77].
Detection of critical points. Any critical point (x∗ , λ∗ ) is characterized
by the fact that Fx (x∗ , λ∗ ) is singular, which will show up as a convergence
failure of the local Newton method for iterates sufficiently close to (x∗ , λ∗ ). As
a consequence, turning points with respect to λ are safely detected: beyond
these points, the local Newton iteration will repeatedly activate the termination criterion (5.21). Generally speaking, the feasible stepsizes Δλmax will
shrink as (x, λ) → (x∗ , λ∗ ). On the other hand, due to [Δλmax ] ≥ Δλmax ,
an equivalent behavior for the computational estimates cannot be guaranteed: since these estimates are only based on pointwise sampling of F and F possible ‘jumps beyond critical points without notice’ cannot be excluded.
Fortunately, extensive computational experience has demonstrated that the
stepsize control derived herein is quite sensitive, typically exhibiting marked
stepsize reductions in the neighborhood of critical points. Summarizing, critical points of order k > 0 are rather often, but not safely detected in continuation methods with explicit parametrization.
Jacobian ill-conditioning. When approaching a critical point on the homotopy path, the Jacobian condition number is known to increase, which might
support the idea of estimating it along the continuation process. In connection with QR-decomposition, the subcondition number sc(Fx ) is cheaply at
hand [83]. Further condition number estimates may be found within Matlab.
In the experience of the author, the most reliable technique for general linear
equations has been found to be based on iterative refinement with the same
mantissa length (cf. I. Jankowsky/H. Woźniakowski [122]): Let δx denote the
correction computed from the linear residual equation and let ε denote the
relative machine precision. Then a rough estimate of the condition number is
δx∞
.
cond (Fx ) = cd (Fx ) =
.
εx∞
Bibliographical Note. The stepsize control presented here is based on
the author’s habilitation thesis [61], there in the context of optimal control
problems attacked by multiple shooting techniques—compare [71, Section
8.6.2]. In Section 7.1.3 below, the success of these methods is documented at
a space shuttle problem (Example 7.1).
250
5 Parameter Dependent Systems: Continuation Methods
Explicit reparametrization beyond turning points. The mathematical
reason, why turning points with respect to the parameter λ cannot be computed via Newton continuation is that in the neighborhood of these points
the parametrization with respect to λ breaks down. In order to be able to
pass beyond turning points, W.C. Rheinboldt and J.V. Burkardt [178] suggested a technique that selects any of the components x1 , . . . , xn+1 (identifying λ = xn+1 ) for local parametrization, if only the curve can be locally
parametrized by that component. This approach is rather popular in computational science and engineering, whenever the basic structure of a bifurcation
diagram is known from insight into the problem at hand.
In the absence of such insight, explicit parametrization can also be automated: the selection may be based on the occurrence of ‘small’ pivots within
Jacobian LU -decompositions during the Newton continuation process. If QRdecomposition with column permutation is realized, then the last column will
be selected. Apart from the choice of a corresponding single component of x,
any norm of x can be chosen likewise. The selected artificial parameter will
then be used for discrete continuation, while the local Newton method runs
over the remaining n components (code PITCON, which realizes the strategy
of [178]).
Note, however, that this approach requires some switching between components of x, which introduces an element of nondifferentiability into the
algorithm and, as a consequence, may cause some lack of robustness.
5.2 Gauss-Newton Continuation Method
In this section we again study the numerical solution of parameter dependent
systems of nonlinear equations
F (y) = 0
in terms of the extended variable y = (x, λ), x ∈ Rn , λ ∈ R1 . In contrast
to Section 5.1, however, local Newton methods are now replaced by local
Gauss-Newton methods (see Section 4.4.1), which open the possibility to
some smooth pathfollowing beyond turning points—as will be shown next.
5.2.1 Discrete tangent continuation beyond turning points
Throughout the present section we assume that a numerical solution y ∗ =
(x∗ , λ∗ ) is at hand—either gained directly from insight into the problem or
computed via a local or global Gauss-Newton method for underdetermined
equations (see Section 4.4). As already discussed earlier, Newton methods
with explicit λ-parameterization are bound to fail in the neighborhood of
turning points, since there this parameterization breaks down.
5.2 Gauss-Newton Continuation Method
251
Pseudo-arclength continuation. As a first idea to overcome this difficulty
we may resort to differential geometry, where the smooth parametrization
with respect to the arclength s is usually recommended: in addition to the n
equations
F (y(s)) = 0
this parameterization includes the normalizing condition
2 2 2
dy = dx + dλ
= 1,
ds ds ds
2
2
which after discretization eventually leads to the pseudo-arclength parametrization
Δy22 − Δs2 = 0
as normalizing condition.
Bibliographical Note. The idea of pseudo-arclength continuation has
been suggested and worked out in first details by H.B. Keller in [130]. Its most
mature and popular implementation is in the code AUTO due to E. Doedel [89],
a code known to be rather robust and reliable. Since a sound theoretically
backed control of the above stepsize parameter Δs is hard to design, the code
realizes some empirical stepsize control. From the invariance point of view,
the concept of arclength is not even invariant under rescaling of the parameter
λ—see Exercise 5.1, where an interesting limiting case is discussed.
Gauss-Newton continuation idea. Here we will follow an alternative
idea: at any point y ∈ Rn+1 including turning points, the local tangent is
well-defined via the underdetermined system of equations
F (y)t(y) = 0 .
As long as rank F (y) = n, the mapping t(y) is known to vary smoothly along
the parameter curve also beyond turning points. Suppose now we parameterize the curve y locally with respect to some coordinate s > 0 along the
unique tangent direction t starting at the previous solution point y ν = y(0).
Continuation along t then defines some prediction path (for ν = 0, 1, . . .)
yν+1 (s) = y ν + sν t(y ν ),
sν > 0 .
(5.26)
The prediction path supplies possible starting points y 0 = yν+1 for the local
quasi-Gauss-Newton iteration towards the next solution point y ν+1 .
As derived in Section 4.4.1, we may construct a quasi-Gauss-Newton method,
equivalent to a local quasi-Newton method in the n-dimensional hyperplane
H := y(s) ⊕ N ⊥ F (
y (s)) .
252
5 Parameter Dependent Systems: Continuation Methods
t(y(s))
y(s)
t(ŷ(s))
ŷ(s)
y(0)
s · y(0)
H
Fig. 5.4. Discrete tangent continuation in y = (x, λ).
The geometric situation is represented in Figure 5.4. Observe that y ν+1 is just
defined as the intersection of H with the solution curve. A natural coordinate
frame for this setting is y = y(s) + (u, σ), u ∈ H(s) = Rn , σ ∈ R1 .
In view of the straightforward estimate
yν (s) − yν (s) ≤
1
max ÿν (δ)s2
2 δ∈[0,s]
(5.27)
the discrete tangent continuation method is seen to be of order 2—compare
(5.7) and definition (5.6). In order to implement the actual order p = 2,
tangent continuation requires a sufficiently accurate approximation of the
Jacobian both at each solution point y ν and at each starting point yν .
Tangent computation via QR-decomposition. Assume that we realize the rank-deficient pseudoinverse J + of the Jacobian (n, n + 1)-matrix J
through the QR-Cholesky algorithm (4.77) as given in Section 4.4.1. Then,
using the vector w = R−1 S and the permutation Π as defined therein, we
can compute the normalized kernel vector t as
*
−w
t := ±Π
1 + wT w .
1
One method of fixing the arbitrary sign is to require the last component of
t(y ν ), say tξ , to have the same sign as ξν − ξ ν−1 —thus defining a natural
orientation also around turning points.
Tangent computation via LU -decomposition. In large sparse problems
a direct sparse solver based on LU -decomposition will be applied. During the
actual decomposition of the (n, n + 1)-matrix J, the pivoting strategy with
possible column permutations Π will give rise to a zero pivot. Upon dropping
the associated column of J and setting the corresponding component to zero,
say z̃ξ = 0, a particular solution z̃ satisfying
5.2 Gauss-Newton Continuation Method
J z̃ = −F
253
(5.28)
can be computed. In order to solve Jt = 0 for some (unnormalized) kernel
vector t, we may set the component of t associated with the zero pivot column
to some nonzero value, say tξ = 1. Upon using the relations
t tT
+
+
z := −J F = J J z̃ = I − T
z̃
t t
we arrive at the computationally attractive representation
z = z̃ −
(t, z̃)
t
(t, t)
(5.29)
in terms of the Euclidean inner product (·, ·).
Computation of quasi-Gauss-Newton corrections. The actual computation of the quasi-Gauss-Newton corrections
Δy k = −Jk+ F (y k )
requires the computation of the Moore-Penrose pseudo-inverse of the Jacobian updates Jk . Since this quasi-Gauss-Newton update preserves the
nullspace component, we can use a simple variant of the recursive quasiNewton method, given as algorithm QNERR in Section 2.1.4: with J0 = F (y 0 ),
we only need to formally replace J0−1 by J0+ , wherever this term arises.
5.2.2 Affine covariant feasible stepsizes
In order to develop an adaptive stepsize control (see Section 5.2.3 below),
theoretical feasible stepsizes are studied first. As in the case of the Newton
continuation method, an affine covariant setting appears to be natural, since
the path concept is the dominating one in continuation. Since the local quasiGauss-Newton iteration has been shown to be equivalent to a quasi-Newton
iteration in H (see Figure 5.4), we may proceed as in the simpler Newton
continuation method (Section 5.1.2) and model the situation just by the
simplified Newton iteration in H.
Theorem 5.7 Consider the discrete tangent continuation method (5.26) in
combination with the simplified local Gauss-Newton iteration starting at
y 0 := y(s) = yν and with Jacobian approximations Jk := F (y 0 ) for k ≥ 0.
Let F : D ⊂ Rn+1 → Rn denote some C 1 -mapping with D open, convex,
and sufficiently large. Then, under the assumption of the affine covariant
Lipschitz conditions
0 + F (y ) F (y) − F (y 0 ) ≤ ωH y − y 0 , y, y 0 ∈ H
254
and
5 Parameter Dependent Systems: Continuation Methods
+ F (y) F (u + δ2 t(u), −F (u) t(u) ≤ δ2 ωt t(u), 22
2
with the (normalized) kernel vector t(u) = kerF (u) and
y, u + δ2 t(u) ∈ D , 0 ≤ δ2 ≤ 1 ,
the simplified Gauss-Newton iteration converges for all
√
s ≤ smax := 1/ ωH ωt .
(5.30)
Proof. For the simplified Newton iteration in H we may apply Theorem 2.5,
which here requires the verification of the sufficient condition
Δy 0 (s)ωH ≤ α0 (s)ωH ≤
1
2
.
(5.31)
For simplification, we introduce the notation
J(s) := F (
y (s)) , F (s) := F (
y (s)) , t(s) := t(
y (s))
so that
F (0) = 0 , J(0)t(0) = 0 .
Then the derivation of an appropriate α0 (s) may proceed as follows:
0 Δy (s) = J(s)+ F (s) = J(s)+ F (s) − F (0) = J(s)+
s
δ=0
s
=
J(δ)t(0)dδ ≤
s
J(s)+ J(δ)t(0) dδ
δ=0
J(s)+ J(δ) − J(0) t(0) dδ ≤ 12 ωt s2 .
δ=0
Hence
α0 (s) := 12 ωt s2 ,
(5.32)
which inserted above directly leads to the maximum feasible stepsize smax .
As an extension of the simpler Newton continuation case treated in Section
5.1.2 we next study the ‘movement’ of H along the parameter curve.
Theorem 5.8 Assumptions and notation as in the preceding Theorem 5.7.
Let
y(s) := y(0) + st y(0)
5.2 Gauss-Newton Continuation Method
255
denote a short-hand
notation for the discrete tangent continuation (5.26). Let
t := t(s) = t y(s) , s fixed. Define y(s) as the intersection of H(s) with the
solution curve and let
cs := t(s)T t y(s) .
Then, for cs = 0, one has
2
ÿ(s) ≤ ωt c0 .
2
|cs |3
(5.33)
Proof. For simplification we introduce
z(s) := y(s) − y(s)
so that
ż(s)
z̈(s)
= ẏ(s) − t(0) ,
= ÿ(s) .
(5.34)
(5.35)
For fixed s and t(s) = t(
y ), we obtain
t(s)T z(s) ≡ 0 ,
t(s)T ż(s) ≡ 0 ,
t(s)T z̈(s) ≡ 0
(5.36)
(5.37)
for 0 ≤ s ≤ s. Variation of s defines y(s) by virtue of
F y(s) ≡ 0 ,
which implies
F y(s) ẏ(s) ≡ 0 ,
2
F y(s) ẏ(s) + F (y(s))ÿ(s) ≡ 0 .
(5.38)
Next, from rank F (y) = n for all y ∈ D and (5.38), we may write
ẏ(s) = γ(s)t y(s)
in terms of some coefficient γ to be determined. From this we obtain
z̈(s) = β(s)t y(s) − η(s)
(5.39)
in terms of some coefficient β and some vector
η ⊥ t y(s) .
256
5 Parameter Dependent Systems: Continuation Methods
Upon collecting these relations, we arrive at the expression
+ 5
62
η(s) = F y(s) F y(s) γ(s)t y(s)
.
(5.40)
The determination of β(s), γ(s) starts from
2
z̈(s)2 = η(s)22 + β 2 (s) ,
since t is normalized. Upon combining (5.34) and (5.36) we obtain
0 = tT ż(s) = tT γ(s)t(y(s)) − t(0) ,
from which
γ(s) =
c0
cs
for cs = 0 .
Application of the same procedure with (5.35), (5.39) and (5.37) yields
0 = tT z̈(s) = tT β(s)t(y(s)) − η(s)
or, equivalently
β(s) =
tT η(s)
cs
for cs = 0 .
(5.41)
As t = t y(s) , we may continue
T
t y(s) η(s) = 0 .
Hence
|tT η(s)|
T
= tT η(s) − cs t y(s) η(s) T
= tT η(s) − tT t y(s) t y(s) η(s) = tT I − t(y(s))tT (y(s)) η(s) ≤ I − t(y(s))tT (y(s)) t · η(s)2
2
=
1−
c2s
η(s)2 .
Insertion into (5.41) yields
β(s) ≤
which leads to
1 − c2s η(s) ,
|cs |
2
5.2 Gauss-Newton Continuation Method
2
z̈(s)2 ≤ η(s)22 +
257
1 − c2s
η(s)22
2
η(s)
=
.
2
c2s
c2s
Finally, estimation of η(s) in (5.40) supplies
η(s)2 ≤ ωt (γ(s))2
and, therefore
z̈(s)2 ≤ ωt ·
c0
cs
2
·
1
,
|cs |
which with (5.35) completes the proof.
The result (5.33) shows that the Lipschitz constant ωt just measures the
local curvature ÿ(s) of the parameter curve. Insertion into (5.27) specifies the
second order coefficient. In view of actual computation we may eliminate the
variation of cs via the additional turning angle restriction
c0 = min cδ > 0 ,
δ∈[0,s̄]
which, in turn, yields a stepsize restriction, of course. Then the bound (5.33)
can be replaced by the corresponding expression
ÿ (s) ≤ ωt /c0 .
(5.42)
2
For an affine contravariant derivation of feasible stepsizes see Exercise 5.2.
5.2.3 Adaptive stepsize control
On the basis of the theoretical results above, we are now ready to derive
computational estimates for feasible stepsizes sν according to (5.26). Recall
again that the first quasi-Gauss-Newton step may be interpreted as an ordinary Newton step in H, whereas further quasi-Gauss-Newton steps are just
1
quasi-Newton steps in H. Let Δy denote the first simplified Gauss-Newton
correction, which is cheaply available in the course of the computation of the
first quasi-Gauss-Newton correction. Then a first contraction factor Θ0 must
satisfy
1
Δy Θ0 :=
≤ 12 ωH α0 ,
(5.43)
Δy 0 which with (5.31) implies that
Θ0 ≤ Θ :=
1
4
.
In the spirit of our paradigm in Section 1.2.3, we construct computational
stepsize estimates [·] on the basis of the theoretical stepsizes (5.30) as
258
5 Parameter Dependent Systems: Continuation Methods
[smax ] := 1/ [ωH ] [ωt ] .
(5.44)
As usual, since
[smax ] ≥ smax
both a prediction and a correction strategy need to be designed.
Correction strategy. From (5.43) we may obtain
[ωH ] :=
2Θ0
≤ ωH
Δy 0 (5.45)
and similarly from (5.32)
[ωt ] :=
2 Δy 0 ≤ ωt .
s2
Upon inserting these estimates into (5.44), we are led to the stepsize suggestion
Θ
sν :=
sν .
Θ0
This estimate requires the knowledge of an actual stepsize sν , which means
that it may only serve within a correction strategy. Whenever the termination
criterion (for k ≥ 0)
Θk > 12
holds, then the quasi-Gauss-Newton iteration is terminated and the previous
continuation step (5.26) is repeated supplying the new starting point
yν+1
= y ν + sν t(y ν ) ,
wherein roughly
sν < 0.7sν
is guaranteed.
Prediction strategy. For the construction of a prediction strategy, we may
combine the relations (5.42) and (5.27) to obtain
y ν − yν ≤ 12 ωt s2ν−1
1
.
c0
This naturally defines the computational estimate
[ωt ] :=
2c0 y ν − yν 2
≤ ωt .
s2ν−1
Together with [ωH ] from (5.45), the general formula (5.44) leads to the stepsize suggestion
5.2 Gauss-Newton Continuation Method
s0ν
:=
Δy 0 Θ 1
y ν − yν Θ0 c0
259
1/2
sν−1 .
(5.46)
Clearly, this is a direct extension of the prediction formula (5.25) for the
Newton continuation method. Recalling that g(Θ) ≈ 2Θ, the main new item
appears to be the trigonometric factor
c0 = tT (
yν )t y ν−1 ,
which roughly measures the ‘turning angle’ of the hyperplane Hν−1 ⊃ y ν−1
to Hν ⊃ yν —see again Figure 5.4.
Finally, as in the Newton continuation scheme, precaution must be taken for
the nearly linear case
Θ0 ≤ Θmin 1 ,
in which case Θ0 → Θmin in (5.46).
Detection of critical points. In the described tangent continuation turning
points (as critical points of order k = 0) do not play any exceptional role. The
occurrence of critical points of order k ≥ 1, however, needs to be carefully
monitored. For this purpose, we define by
dλ := det (Fx )
the determinant of the (n, n)-submatrix of the Jacobian F (y) that is obtained
by dropping the λ-column, which is Fx , of course. Similarly, let dξ denote the
determinant of the submatrix, where the last column has been dropped—
which corresponds to the internal parameter ξ, in general different from the
external parameter λ, when column permutations based on pivoting are involved. For the safe detection of critical points, we compute the determinant
pair (dξ , dλ ) along the solution curve. The computation of dξ is easily done via
det(R)—with a possible sign correction, by (−1)n for n Householder reflections or for the actually performed permutations in the LU-decomposition. If
λ = ξ, then the computation of dλ requires the evaluation of the determinant
of a Hessenberg matrix, which means O(n2 ) operations only. Sign changes of
this pair clearly indicate the occurrence of both turning points and simple
(possibly unfolded) bifurcation points—see Figure 5.5 for an illustration of
the typical situations. This device has a high degree of reliability in detecting
turning and simple bifurcation points and a good chance of detecting higher
order critical points. Safety,of course, cannot be guaranteed as long as only
pointwise sampling is used.
Computation of turning points. Assume that the discrete continuation
method has supplied some internal embedding parameter ξ (associated with
the last column in the matrix decomposition) and some interval ξ, ξ supposed to contain a turning point, say ξ ∗ . Then the implicit mapping λ(ξ) will
have a minimum or maximum value within that interval so that
260
5 Parameter Dependent Systems: Continuation Methods
ξ
(+, +)
ξ
(−, +)
(+, +)
(+, −)
τ
ξ
(+, +)
(+, −) τ
(−, −)
(−, −)
ξ
(+, +)
(−, −)
(+, −)
τ
(+, +)
τ
Fig. 5.5. Sign structure of determinant pair (dξ , dλ ). Upper left: turning point,
upper right: detected simple bifurcation and unfoldings, lower left: pair of turning
and bifurcation point or pitchfork bifurcation, lower right: possibly undetected bifurcation point due to too small branch angle.
λ̇(ξ ∗ ) = 0 .
(5.47)
On this basis, the following algorithm for the determination of turning points
is recommended:
(I) Construct the cubic Hermite polynomial p(ξ), ξ ∈ ξ, ξ such that
p ξ =λ ξ , p ξ =λ ξ ,
ṗ ξ = λ̇ ξ , ṗ ξ = λ̇ ξ .
As an approximation of the unknown implicit equation (5.47), we solve
the quadratic equation
ṗ(ξ) = 0 .
(5.48)
The usual bisection assumption
λ̇ ξ λ̇ ξ ≤ 0 ,
(5.49)
then assures that equation (5.48) has a real root ξ ∈ ξ, ξ .
(II) Perform aGauss-Newton
iteration of standard type with starting point
0
y = y ξ = x ξ , ξ . Let y∗ denote the point obtained on the solution curve.
5.2 Gauss-Newton Continuation Method
(III) As soon as
261
y − y∗ ≤ ε
holds for some prescribed (relative) accuracy ε, then y∗ is accepted as
turning point approximation. Otherwise, ξ∗ replaces either ξ or ξ such
that (5.49) holds and step (I) is repeated.
The above algorithm fits into the frame of a class of algorithms, for which
superlinear convergence has been proved by H. Schwetlick [183].
Graphical output. Here both nodal data {y ν } and their local tangents
t({yν }) are usually given in different parametrizations corresponding to the
different internal parameters ξν . As a consequence, Bezier-Hermite splines
appear to be the method of choice for the graphical output—compare, e.g.,
Section 7.3 in the textbook [77] or any book on computer aided design.
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.00
0.02
0.04
0.06
0.08
0.10
Fig. 5.6. Chemical reaction problem: x2 (λ). Crosspoint just by projection.
Example 5.2 Chemical reaction problem. This model due to M. Kubiček
[137] reads (see Figure 5.6):
λ(1 − x3 ) exp(10x1 /(1 + 0.01x1 )) − x3
= 0
22λ(1 − x3 ) exp(10x1 /(1 + 0.01x1 )) − 30x1
= 0
x3 − x4 + λ(1 − x4 ) exp(10x2 /(1 + 0.01x2 ))
= 0
10x1 − 30x2 + 22λ(1 − x4 ) exp(10x2 /(1 + 0.01x2 ))
= 0.
262
5 Parameter Dependent Systems: Continuation Methods
Example 5.3 Aircraft stability problem. In [149] R.G. Melhem and
W.C. Rheinboldt presented the following problem:
−3.933x1 + 0.107x2 + 0.126x3 − 9.99x5 − 45.83λ
−0.727x2x3 + 8.39x3 x4 − 684.4x4x5 + 63.5x4 λ =
−0.987x2 − 22.95x4 − 28.37u + 0.949x1 x3 + 0.173x1 x5
0
=
0
−1.578x1 x4 + 1.132x4 λ =
0
0.002x1 − 0.235x3 + 5.67x5 − 0.921λ − 0.713x1 x2
x2 − x4 − 0.168u − x1 x5
=
0
−x3 − 0.196x5 − 0.0071λ + x1 x4
=
0.
Herein x1 , x2 , x3 are the roll rate, pitch rate, and yaw rate, respectively, x4 is
the incremental angle of attack, and x5 the sideslip angle. The variable u is
the control for the elevator, λ the one for the aileron. The rudder deflection
is set to zero. For u = 0 this problem is symmetric: F (x, λ) = F (x, −λ). For
u = −0.008 the perturbed symmetry is still visible, see Figure 5.7.
0.08
0.04
0.00
-0.04
-0.08
-0.12
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Fig. 5.7. Aircraft stability problem: x4 (λ), perturbed symmetry.
Bibliographical Note. The adaptive pathfollowing algorithm, as worked
out here, has been implemented in the code ALCON1 due to P. Deuflhard,
B. Fiedler, and P. Kunkel [72].
5.3 Computation of Simple Bifurcations
263
5.3 Computation of Simple Bifurcations
Suppose that the numerical pathfollowing procedure described in Section
5.2 has produced some guess y 0 of an expected close-by simple bifurcation
point y ∗ —see, e.g., the double determinant detection device presented in Section 5.2.3. Then the task is to either compute a bifurcation point y ∗ iteratively
from the starting point y 0 or to decide that there is none in the neighborhood of y 0 . In Section 5.3.1, we will first study the basic construction of
augmented systems that have certain critical points of order k > 0 as locally
unique solutions—excluding turning points (k = 0), which can be computed
easier as shown in the preceding section. The general construction scheme
for augmented systems will be based on the theory of universal unfolding of
singularities, which in the case of simple bifurcations specifies to the system
of G. Moore. In Section 5.3.2, certain Newton-like algorithms for an efficient
solution of that augmented system will be worked out in some detail. On the
basis of structure preserving block elimination techniques for each Newton
step, details of the branching-off algorithm are elaborated in Section 5.3.3—
involving the computation of entering and emanating semi-branches as well
as the restart of discrete tangent continuation on the new semi-branches.
5.3.1 Augmented systems for critical points
Let y ∗ denote a perfect or unperturbed singularity of order k ≥ 1 with
F (y ∗ ) = 0
and
rank F (y ∗ ) = n − k .
(5.50)
Even though we will later only work out an algorithm for simple bifurcations
(k = 1), we include the more general case k > 1 here as well—to make the
general construction of augmented systems transparent.
Lyapunov-Schmidt reduction. In the notation from above, let A :=
F (y ∗ ), N (A) = ker(A) its (k + 1)-dimensional nullspace, and R⊥ (A)
its k-dimensional corange. If we again introduce the orthogonal projectors
⊥
P := A+ A , P := AA+ , we have that P ⊥ projects onto N (A) and P onto
⊥
R (A). With this notation we may define the natural splitting
y = y∗ + v + w ,
w := P (y − y ∗ ) ,
v := P ⊥ (y − y ∗ )
in the space of the unknowns. From assumption (5.50) and the implicit function theorem, we know that there exists a function w∗ such that
P F (y ∗ + v + w) = 0 ⇐⇒ w = w∗ (v) .
264
5 Parameter Dependent Systems: Continuation Methods
Replacement of the variable w by the function w∗ then leads to a reduced
system of k equations in k + 1 unknowns
⊥
f (v) := P F (y ∗ + v + w∗ (v)) = 0 .
(5.51)
This is the well-known Lyapunov-Schmidt reduction, which stands at the
beginning of every mathematical treatment of singularities—see, e.g., the
classical book edited by P.H. Rabinowitz [173]. For actual computation we
need to define orthogonal bases for both N and R⊥ as
R⊥ (A) =: z1 , . . . , zk N (A) =: t1 , . . . , tk+1 or, in equivalent matrix notation, as
t := [t1 , . . . , tk+1 ] , z := [z1 , . . . , zk ] ,
so that
At = 0,
AT z = 0,
tT t = Ik+1 , P ⊥ = ttT ,
z T z = Ik , P
⊥
:= zz T .
(5.52)
Note that t and z are only specified up to orthogonal transformations—which
leaves dim O(k + 1) = 12 k(k + 1) degrees of freedom for t and dim O(k) =
1
k(k − 1) degrees of freedom for z.
2
Upon introducing local coordinates ξ ∈ Rk+1 , γ ∈ Rk by virtue of
k+1
v=
k
ξi ti = tξ ,
f (v) =
i=1
γj zj = zγ ,
j=1
the reduced equations (5.51) can be rewritten in the form
γ(ξ) := z T f (tξ) = z T F (y ∗ + tξ + w∗ (tξ)) = 0 .
(5.53)
For the actual determination of singularities, higher order derivatives of both
sides will play an important role.
Lemma 5.9 Assumptions as just introduced. Let y ∗ := 0 for convenience
and ai ∈ Rk+1 . Then the following relations hold:
γ̇(0)a1
=
0
(5.54)
γ̈(0)[a1 , a2 ] = z T F [ta1 , ta2 ]
(5.55)
...
γ (0)[a1 , a2 , a3 ] = z T F [ta1 , ta2 , ta3 ]
−z t F [ta1 , A+ F [ta2 , ta3 ]]
−z F [ta2 , A F [ta3 , ta1 ]]
−z T F [ta3 , A+ F [ta1 , ta2 ]]
T
+
(5.56)
5.3 Computation of Simple Bifurcations
265
Proof. As a consequence of y ∗ = 0 we have v∗ = 0, w∗ (0) = 0 and ξ ∗ = 0.
We start from
γ(ξ) = z T F (tξ + w∗ (tξ))
(5.57)
and
P F (tξ + w∗ (tξ)) ≡ 0 .
(5.58)
Differentiation of (5.57) with respect to ξ yields
γ̇(ξ)a1 = z T F (tξ + w∗ (tξ))(ta1 + wξ∗ (tξ)a1 )
and, after insertion of z T A = 0
γ̇(0)a1 = z T A(ta1 + wξ∗ (0)a1 ) = 0 ,
which confirms (5.54). Differentiation of (5.58) yields
P A(ta1 + wξ∗ (0)a1 ) = 0 .
which, with P A = A, At = 0 and wξ∗ (0)a1 ∈ N ⊥ , can be solved by
wξ∗ (0)a1 = 0 .
(5.59)
Upon differentiating (5.57) once more and inserting the expression above we
obtain
γ̈(0)[a1 , a2 [= z T F (0)[ta1 , ta2 ] ,
which confirms (5.55). Differentiation of (5.58) once more leads to
∗
P F (0)[ta1 , ta2 ] + P Awξξ
(0)[a1 , a2 ] = 0 .
With arguments as just used before, the latter equation can be solved to yield
∗
wξξ
(0)[a1 , a2 ] = −A+ F (0)[ta1 , ta2 ] .
(5.60)
Upon differentiating (5.57) for a third time, we eventually arrive at
···
γ (0)[a1 , a2 , a3 ] =
z T F (0)[ta1 , ta2 , ta3 ]
∗
+z T F (0)[ta1 , wξξ
(0)[a2 , a3 ]]
∗
+z T F (0)[ta2 , wξξ
(0)[a3 , a1 ]]
∗
+z T F (0)[ta3 , wξξ
(0)[a1 , a2 ]] .
∗
Insertion of wξξ
(0) then finally confirms (5.56).
266
5 Parameter Dependent Systems: Continuation Methods
Universal unfolding. It is clear from the above derivation that the function
γ : Rk+1 → Rk contains all essential information needed to classify the local
structure of a singularity. Since only derivatives of γ up to a certain order are
required, it suffices to study simple polynomial germs g(ξ), which then may
stand for the whole function class
Γ (g) := {γ(ξ) = β(ξ)g(h(ξ)) | β, h C ∞ -diffeomorphism, h(0) = 0}
called the contact equivalence class. From a geometrical point of view all
germs within one equivalence class show a similar solution structure around
ξ = 0. We may say that the reduced mapping γ is contact equivalent to a
representative germ g ∈ Γ (γ) or, vice versa, γ ∈ Γ (g). As examples, the germ
gs (ξ) := ξ12 − ξ22
(k = 1)
represents a simple bifurcation, whereas the germ
gc (ξ) := ξ12 − ξ23
(k = 1)
characterizes an asymmetric cusp. We then obtain
β(ξ)g(h(ξ)) = z T F (y ∗ + tξ + w∗ (tξ)) = 0
in terms of certain diffeomorphisms β, h.
Up to now, our analytical presentation has only covered perfect singularities.
An efficient algorithm will have to deal with imperfect or unfolded singularities
y ∗ as well—even without knowing in advance about the structure of the
perturbations. As an immediate consequence, we will encounter γ(0) = 0
and the associated Jacobian matrix F (y ∗ ) may no longer be exactly rankdeficient, but still ‘close to’ a rank-deficient matrix. In this situation, the
structure of topological perturbations is important, which are known to lead
to an unfolding of nongeneric singularities. In the just introduced framework,
such perturbations may be written as polynomial perturbations p(ξ, α) of the
germs g replacing them by the perturbed germs
G(ξ, α) := g(ξ) + p(ξ, α) ,
(5.61)
wherein p(ξ, 0) ≡ 0 and the parameters α denote the unfolding parameters.
The minimal number of unfolding parameters is a characteristic of each type
of singularity and is called its codimension q. In case of this minimal parameterization, the representation (5.61) is called universal unfolding. A special
feature of any universal unfolding is that monomials arising in p are at least
2 orders less than corresponding monomials arising in g. As examples again,
gs for the simple bifurcation is replaced by
Gs (ξ, α) := ξ12 − ξ22 + α ,
(k = 1, q = 1) ,
whereas gc for the asymmetric cusp is replaced by
Gc (ξ, α1 , α2 ) := ξ12 − ξ23 + α1 + α2 ξ2 ,
(k = 1, q = 2) .
(5.62)
5.3 Computation of Simple Bifurcations
267
Bibliographical Note. For a general thorough treatment of unfolded
singularities, the reader may refer to M. Golubitsky and D. Schaeffer [109]
and their textbooks [110, 111]. Since these authors treat dynamical systems
ẋ = F (x, λ) with state variables x, they give the explicit parameter λ an
extra role—in contrast to the present section here, which treats all n + 1
components of y = (x, λ) the same.
Construction of augmented systems. Summarizing, we may assume that
specific diffeomorphisms β, h exist such that
β(ξ)G(h(ξ), α) = z T F (y ∗ + tξ + w∗ (tξ))
(5.63)
with, in general,
G(h(0), α) = G(0, α) = p(0, α) = 0 .
Therefore, for perturbed singularities, the reduced system (5.53) must be
replaced by
z T F (y ∗ ) = p(0, α) .
These k equations, together with the n − k equations P F = 0, then lead to
the n equations
F (y ∗ ) = z p(0, α)
in terms of the (k + 1)n + q + 1 unknowns (y, z, α).
Simple bifurcation. Let us now return to the special case k = 1, q = 1.
Here we arrive at the augmented system of G. Moore [103]:
F (y)T z = 0 ,
(5.64)
F (y) + αz = 0 ,
(5.65)
− 1) = 0 .
(5.66)
1 T
(z z
2
It comprises (2n + 2) nonlinear equations for the (2n + 2) unknowns (y, z, α).
The Jacobian J(y, z, α) of this mapping is nonsingular for sufficiently small
perturbation parameter α. The proof of this fact is postponed to Section 5.3.2
below, since it stimulates an algorithmic idea for the iterative solution of the
above augmented system.
In order to make sure that a geometrical bifurcation really exists locally, we
must impose the additional second derivative condition
z T F (y ∗ )[t, t]
nondegenerate, indefinite .
(5.67)
This condition assures the existence of two local branch directions as will
be shown next. In most of the established analysis treatise—compare, e.g.,
M.G. Crandall and P.H. Rabinowitz [46]—one of the intersecting branches is
268
5 Parameter Dependent Systems: Continuation Methods
assumed to be the trivial one—which is acceptable for a merely analytical
treatment, but unsatisfactory for the construction of efficient numerical algorithms. Therefore we here include a theorem that treats both branches as
indistinguishable.
Theorem 5.10 Assumptions and notation as just introduced. Let F ∈ C k ,
k ≥ 3. Then, in a neighborhood of y ∗ , the solution set F + αz = 0 consists of
two one-dimensional C k−2 -branches γ1 (s), γ2 (s) such that
γi (0) = y ∗ , i = 1, 2 ,
N = γ̇1 (0), γ̇2 (0) ,
5
6
z T F (y ∗ ) γ̇i (0), γ̇i (0) = 0 .
Proof. (Sketch) For convenience, assume again that y ∗ = 0. Let a standard
Lyapunov-Schmidt reduction have been performed in terms of a parametrization of the two-dimensional nullspace N . It is then sufficient to study
⊥
the mapping F = P F : N → R. Introducing polar coordinates, define a
blow-up version of F by
⎧
*
⎪
r2
r>0
⎨ 2F r e(Θ)
Φ(r, Θ) :=
5
6
⎪
⎩ F (0) e(Θ), e(Θ) r = 0 ,
where e(Θ) := (cos Θ, sin Θ) denotes the unit vector in the direction
Θ ∈ [0, π]. Obviously, Φ = 0 holds if F = 0. Moreover, F ∈ C k implies
Φ ∈ C k−2 and Φ can also be formally continued to r < 0. By assumption
(5.67) there exist directions Θi (i = 1, 2) such that
Φ(0, ±Θi )
=
ΦΘ (0, ±Θi ) =
0,
5
6
F (0) eΘ (Θi ), eΘ (Θi ) = 0 .
Hence, by the implicit function theorem, there are four C k−2 –semi-branches
γi± (r) = r e(Θi± (r)) , i = 1, 2 r ≥ 0
for sufficiently small r. At r = 0, the functions γi+ (r) and γi− (−r) have the
same derivatives up to order k − 2. Therefore, combining γi+(r) and γi− (−r),
i = 1, 2, yields two C k−2 -branches, which completes the proof.
From this result, the desired local tangent directions
t∗i = γ̇i (y ∗ ) ,
i = 1, 2 ,
are seen to be defined from the quadratic equation
z T F (y ∗ ) [t∗i , t∗i ] = 0 ,
which, under the assumption (5.67) has the two distinct real roots t∗1 , t∗2 .
5.3 Computation of Simple Bifurcations
269
Asymmetric cusp. This possibly unfolded singularity has also rank-deficiency k = 1, but codimension q = 2. Recall the perturbed germ Gc = G
from (5.62) to derive the characterization (with h(0) = 0):
G( 0, α)
= α1 ,
(5.68)
Gξ1 (0, α)
= 0,
Gξ2 (0, α)
= α2 ,
(5.69)
= 0,
= 0.
(5.70)
Gξ2 ξ2 (0, α)
Gξ1 ξ2 (0, α)
Note that all nonvanishing derivatives beyond (5.69) and (5.70) such as
Gξ1 ξ1 (0, α) = 2
do not show up, since they are arbitrary due to the arbitrary C ∞ -diffeomorphic transformation β(ξ) in (5.63). Upon differentiating the right-hand side
of (5.63) with h(ξ) = ξ and β(ξ) = 1, we may obtain (as in the simple
bifurcation)
z T F (y ∗ ) = α1
from (5.68), which leads to
F (y ∗ ) = α1 z .
(5.71)
From (5.69) and (5.59) we may verify that
F (y ∗ )t1 = 0 , F (y ∗ )t2 = α2 z .
Finally, from (5.70), (5.60) and z T P
⊥
(5.72)
= z T we arrive at
z T F (y ∗ )[t2 , t2 ] =
0,
z T F (y ∗ )[t1 , t2 ] =
0.
Of course, we will add the perturbed corange condition
F (y ∗ )T z = α2 t2
(5.73)
to be compatible with (5.72). For normalization we will choose the four equations
z T z = 1 , tT1 t1 = tT2 t2 = 1 , tT1 t2 = 0 .
(5.74)
Upon combining (5.71) up to (5.74) we would arrive at an overdetermined
system, 4n + 8 equations in 4n + 5 unknowns. Careful examination for general h(ξ) (with still β(ξ) = 1 w.l.o.g) leads to a replacement of (5.72) and
(5.73) by the perturbed Lyapunov-Schmidt reduction (originally suggested by
R. Menzel)
270
5 Parameter Dependent Systems: Continuation Methods
F (y ∗ )T z
= γ11 t1 + γ21 t2 ,
F (y ∗ )t1
= β11 z ,
∗
F (y )t2
(5.75)
= β21 z .
Thus we end up with the following augmented system for the asymmetric
cusp
F (y) = αz ,
F (y)T z = γ11 t1 + γ21 t2 ,
F (y)t1 = β11 z1 ,
F (y)t2 = β21 z2 ,
z T F (y)[t2 , t2 ] = 0 ,
z T F (y)[t1 , t2 ] = 0 ,
zT z = 1 ,
tT1 t1 = tT2 t2 = 1,
tT1 t2 = 0 .
This system comprises 4n + 7 equations in the 4n + 8 unknowns y, z, t1 , t2 ,
α1 , γ11 , γ21 , β11 , β21 —which means that the system is underdetermined.
The associated augmented Jacobian can be shown to have full row rank for
sufficiently small perturbation parameters α1 , γ11 , γ21 , β11 , β21 .
Higher order critical points. The perturbed system (5.75) is a special case
of the general perturbed Lyapunov-Schmidt reduction, wherein A = F (y ∗ )
has rank-deficiency k > 0:
AT zj
Ati
=
=
k+1
i=1
k
γji ti ,
βij zj ,
j=1
(5.76)
zj zl
= δj,l ,
l ≤ j = 1, . . . , k ,
ti tm
= δi,m ,
m ≤ i = 1, . . . , k + 1 .
This underdetermined system comprises (2k + 1)(n + k + 1) − k 2 equations in
(2k + 1)(n+k + 1) unknowns. It has been suggested by P. Kunkel in his thesis
[138, 139, 140] and worked out using tree structures in [141]. The extended
Jacobian has full row rank at a perfect singularity, where
∗
∗
βij
= γij
= 0.
Note that, also for an imperfect singularity y ∗ , we can verify that
γij = βij = zjT Ati ,
i = 1, . . . k,
j = 1, . . . , k + 1 .
5.3 Computation of Simple Bifurcations
271
However, had we identified γij = βij from the start, then the system would
no longer be uniquely solvable.
The missing k 2 degrees of freedom come from 12 k(k − 1) arbitrary degrees in
z and 12 k(k + 1) arbitrary degrees in t due to orthogonal transformation—
compare (5.76) and (5.52) and the discussion thereafter.
Generally speaking, for higher order singularities the number of equations and
of dependent variables does not agree as nicely as in the simple bifurcation
case. Consequently, quite complicated augmented systems may arise, usually
underdetermined as in the cusp case. Part of such systems are still under
investigation including the problem of their automatic generation by means
of computer algebra systems—see, e.g., D. Armbruster [6]. In principle, a
general bifurcation algorithm would need to represent a whole hierarchy of
augmented systems—which, however, will be limited for obvious reasons.
5.3.2 Newton-like algorithm for simple bifurcations
We return to the augmented system (5.64) of G. Moore. The associated extended Jacobian has the block structure
⎡
⎤
C AT 0
J(y, z, α) := ⎣ A αIn z ⎦
0 zT 0
in terms of the submatrices
C := F (y)T z =
n
fi (y)zi , A := F (y) .
i=1
Note that C and therefore J are symmetric matrices.
Theorem 5.11 At a simple (possibly perturbed) bifurcation point y ∗ with
sufficiently small perturbation parameter α∗ the extended Jacobian J(y ∗ , z ∗ , α∗ )
is nonsingular.
Proof. First J(y, z, 0) is shown to be nonsingular. We start with applying
the singular value decomposition
. /
T 0
A=U V ,
=
,
0 0
U
: orthogonal (n, n)-matrix,
V T : orthogonal (n + 1, n + 1)-matrix,
= diag (σ1 , . . . , σn−1 ) , σi > 0 .
Inserting this decomposition into J(y, z, 0) yields after proper transformation
272
5 Parameter Dependent Systems: Continuation Methods
⎡
C
J=⎣ A
0
AT
0
zT
⎤
⎡
0
C
z ⎦→⎣
0
0
T
0
zT
⎤
0
z ⎦ =: J
0
T
with z := U T z, C := V CV T = C . In the above notation, the equations
F (y)T z = 0 now read
ΣT z = 0 ,
which implies that z T = (0, . . . , 0, 1). If, in addition, we introduce the corresponding partitioning
,
C 11 C 12
T
C=
, C 22 = C 22 (2, 2)-matrix ,
T
C 12 C 22
then
⎡
C 11
⎢ C 12
⎢ J =⎢
⎢
⎣ 0
0
Hence
C 12
C 22
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
⎤
⎥
⎥
⎥.
⎥
⎦
rank(J) = rank J = 2n + rank C 22 .
Upon recalling assumption (5.67), we obtain here
z T F [t, t] = C 22 ,
which assures that C 22 is certainly nonsingular and
rank J(y ∗ , z ∗ , 0) = 2n + 2 .
Finally, by the usual perturbation lemma for symmetric matrices with symmetric perturbation, J(y ∗ , z ∗ , α∗ ) is nonsingular for α∗ ‘sufficiently small’,
which completes the proof.
The above Theorem 5.11 assures that the ordinary Newton method (dropping
the iteration index)
⎡
⎤ ⎡
⎤
⎡
⎤
C AT 0
Δy
F Tz
⎣ A αI z ⎦ ⎣ Δz ⎦ = − ⎣ F + αz ⎦
1 T
0
zT
0
Δα
(z z − 1)
2
is well-defined in a neighborhood of the simple bifurcation point. In the special
situation a Newton-like method characterized by the replacement
5.3 Computation of Simple Bifurcations
273
J(y, z, α) → J(y, z, 0) ,
≈ F (y ∗ )
A = F (y) → A
seems to be preferable, since the associated linear block system (again dropping the iteration index)
⎡
T
C A
⎣ A
0
0 zT
⎤ ⎡
⎤
⎡
0
Δy
z ⎦ ⎣ Δz ⎦ = − ⎣
Δα
0
⎤
AT z
F + αz ⎦
1 T
(z z − 1)
2
(5.77)
is easier to solve. On the basis of the theory of Section 2.1, the thus defined iteration will converge superlinearly for perfect bifurcations (α∗ = 0),
but only linearly for imperfect bifurcations. Since excellent starting points
are available in the present setting, one may even keep the initial Jacobian approximation—thus implementing a variant of the simplified Newton
method. This method permits even further computational savings per iteration step.
Distinction between perfect and imperfect bifurcations. Whenever
a perfect bifurcation arises, then the Newton-like iterates {αk } will approach
zero superlinearly so that the criterion
k+1 1 k α
≤ α 4
will be passed. Otherwise, leading digits of α∗ will show up.
In order to compute the two branch directions t∗1 , t∗2 easily, the above extended
Jacobian matrix needs to be decomposed in some structure preserving way.
We work out two possibilities.
Implementation based on QR-decomposition. Let A = F (y) denote
the Jacobian (n, n + 1)-matrix to be decomposed according to
,
A=Q
R
0
S
εT
-
Q : orthogonal (n, n)-matrix,
Π : permutation (n + 1, n + 1)-matrix,
Π T , where R : upper triangular (n − 1, n − 1)-matrix,
S : (n − 1, 2)-matrix,
ε : 2-vector.
For y ‘close to’ y ∗ , R will be nonsingular and ε ‘small’. Hence, the approximation
.
/
R S
A := Q
ΠT
0 0
will be appropriate within a Newton-like iteration.
Starting points for Newton-like iteration. A starting guess y 0 is available from
the path-following procedure, typically from linear interpolation between the
274
5 Parameter Dependent Systems: Continuation Methods
two points y 1 , y2 on the solution curve (see Figure 5.8) that had activated the
device for the detection of critical points (see Figure 5.5). A starting guess
z 0 can be obtained from solving
T z = 0 ,
A
z22 = 1 ,
which, upon inserting the QR-decomposition directly leads to
z 0 = Qen , en = (0, . . . , 0, 1) .
From this, a natural choice of α0 can be seen to be
α0 = − QT F (y 0 ) .
n
Block elimination. Insertion of the above QR-decomposition into the block
system (5.77) suggests the following partitioning
C11 C12
:= Π T CΠ =
, C22 : (2, 2)-matrix ,
C
C12 C22
w
z := QT z =
,
w ∈ Rn−1 , ζ ∈ R ,
ζ
Δu
Π T Δy =
,
Δu ∈ Rn−1 , Δv ∈ R2 ,
Δv
Δw
T
Δz = Q Δz =
,
Δw ∈ Rn+1 , Δζ ∈ R ,
Δζ
f1
T T
Π A z=
,
f1 ∈ Rn−1 , f2 ∈ R2
f2
g1
QT (F + αz) =
,
g1 ∈ Rn−1 , g2 ∈ R
g2
h := 12 (z T z − 1) .
In this notation, (5.77) now reads
⎡
C
C12 RT 0
⎢ 11
⎢ T
⎢ C12 C22 S T 0
⎢
⎢
⎢ R
S
0 0
⎢
⎢
⎢ 0
0
0 0
⎣
0
0
wT ζ
⎤ ⎡
0
⎥
⎥
0 ⎥
⎥
⎥
w ⎥
⎥
⎥
ζ ⎥
⎦
0
⎤
Δu
⎡
⎢
⎢
⎥
⎢
⎢
⎥
⎢
⎢ Δv ⎥
⎢
⎥
⎢
⎢
⎥
⎢
⎢ Δw ⎥ = − ⎢
⎢
⎥
⎢
⎢
⎥
⎢
⎢ Δζ ⎥
⎢
⎣
⎦
⎣
Δα
⎤
f1
⎥
⎥
f2 ⎥
⎥
⎥
g1 ⎥ .
⎥
⎥
g2 ⎥
⎦
h
Simplified Newton iteration. The initial guess z 0 is equivalent to
w0 = 0 , ζ 0 = 1 ,
5.3 Computation of Simple Bifurcations
275
which may be used to decouple the last two equations. In this simplified
Newton method the last two equations then yield (dropping the index k):
Δα = −g2 ,
Δζ = −h .
In order to solve the remaining three equations, we compute once
RS
C 22
=
S,
:=
C22 −
(5.78)
T
(C12
S
T
T
+ S C12 ) + S C11 S
(5.79)
and repeatedly for each new right-hand side
Rg 1
= g1 ,
C 22 Δv
T
T
T
= −f2 + C12
− S C11 g 1 + S f1 ,
Δu
RT Δw
= −g1 − SΔv ,
= −f1 − C11 Δu − C12 Δv .
Note that the symmetric (2, 2)-matrix C 22 is just the one used in the proof
of Theorem 5.11—which means that C 22 may be assumed to be nonsingular
in a neighborhood of a simple bifurcation point. If it appears to be singular
when decomposed, then there will be no bifurcation point locally. Finally,
back substitution yields
Δu
Δw
Δy = Π
, Δz = Q
.
Δv
Δζ
Implementation based on LU-decomposition. Assume that we start
with the decomposition
.
/
R S
A(ε) := L
ΠT ,
0 εT
where L is a lower triangular matrix obtained from some sparse elimination
technique. Due to conditional pivoting within a sparse solver, say, the entries
ε may be ‘small’. Then A may be replaced according to
:= A(0) .
A
For reasons to be understood below, it is advisable to modify the normalizing
condition z T z = 1 such that
h := 12 (LT z22 − 1) = 0 .
Starting points for Newton-like iteration. With y 0 given, initial guesses z 0 are
easily derived from
276
5 Parameter Dependent Systems: Continuation Methods
T z
A
LT z22
=
=
0,
1.
which, with L = (lij ), leads to
LT z 0
α0
:=
:=
en ,
(F (y 0 ))n /lnn .
Block elimination. The above LU -decomposition is now inserted into the
block system (5.77) with the slight change of normalization as just described.
Δu, Δv,
In comparison with the QR-variant we can keep the notation for C,
f1 , f2 unchanged and introduce the following modified quantities:
w
−1
z := L z =
,
w
∈ Rn−1 , ζ ∈ R ,
ζ
w
T
z := L z =
,
w ∈ Rn−1 , ζ ∈ R ,
ζ
Δw
LT Δz =
,
Δw ∈ Rn+1 , Δζ ∈ R ,
Δζ
g1
−1
L (F + αz) =
, g1 ∈ Rn−1 , g2 ∈ R .
g2
In this notation, (5.77) now reads
⎡
C
C12 RT 0
⎢ 11
⎢ T
⎢ C12 C22 S T 0
⎢
⎢
⎢ R
S
0 0
⎢
⎢
⎢ 0
0
0 0
⎣
0
0
wT ζ
⎤ ⎡
0
⎥
⎥
0 ⎥
⎥
⎥
w
⎥
⎥
⎥
ζ ⎥
⎦
0
⎤
Δu
⎤
⎡
⎢
⎥
⎢
⎢
⎥
⎢
⎢ Δv ⎥
⎢
⎢
⎥
⎢
⎢
⎥
⎢
⎢ Δw ⎥ = − ⎢
⎢
⎥
⎢
⎢
⎥
⎢
⎢ Δζ ⎥
⎢
⎣
⎦
⎣
Δα
f1
⎥
⎥
f2 ⎥
⎥
⎥
g1 ⎥ .
⎥
⎥
g2 ⎥
⎦
h
Simplified Newton iteration. In order to implement this variant, just verify
that the choice of z 0 is again equivalent to
w0 = 0 , ζ 0 = 1 ,
which once more implies that
Δα = −g2 , Δζ = −h .
With S and C 11 from equations (5.78) and (5.79) the remaining system can
be solved as follows
C 22 Δv
= g1 + wΔα
,
T
T
T
= −f2 + C12 − S C11 g 1 + S f1 ,
Δu
RT Δw
= −g1 − SΔv ,
= −f1 − C11 Δu − C12 Δv .
Rg 1
5.3 Computation of Simple Bifurcations
277
Finally, back substitution yields
Δu
Δw
Δy = Π
LT Δz =
.
Δv
Δζ
5.3.3 Branching-off algorithm
Suppose that a simple bifurcation point y ∗ has been computed. Then y ∗ is
the intersection of exactly two solution branches associated with the mapping
F +αz. In order to continue the numerical pathfollowing beyond bifurcations,
one will need to first compute the directions of these branches and second to
design an efficient restart strategy along each new semi-branch.
Computation of branch directions. As described above, the local tangent
directions t∗i , i = 1, 2 are computed from the quadratic equation (5.67). Starting from any of the two presented decompositions, the following parametrization of N is natural:
− S ei
∗
ti := Π
, ei := (cos Θi , sin Θi )
ei
with S as defined by (5.78). In the present notation this equation can be
rewritten as
∗
t∗T
i C ti = 0 ,
which reduces to
eTi C 22 ei = 0
in terms of the symmetric (2, 2)-matrix C 22 known to be nonsingular when
a simple bifurcation point exists locally. This is again a quadratic equation
in either tan Θi or cot Θi with two different real roots under the assumption
(5.67). In case C 22 turns out to be semi-definite or degenerate, then a nonsimple bifurcation point is seen to occur. Complex conjugate roots indicate
an isola (which, however, would be hard to detect by just pathfollowing with
respect to one parameter!).
Stepsize control restart. Suppose we have the situation of one entering
semi-branch (already computed) and three emanating semi-branches (to be
computed next) as depicted in Figure 5.8.
Let, formally, t∗3 := −t∗2 . In order to start the path-following procedure along
each emanating semi-branch, one is required to define starting points y for
the quasi-Gauss-Newton iteration (i = 1, 2, 3):
yi (s) := y ∗ + s · t∗i .
Herein an efficient control of the stepsize s requires some care. Two antagonistic conditions are to be matched:
278
5 Parameter Dependent Systems: Continuation Methods
y∗
y2
y1
Fig. 5.8. Simple bifurcation point: branch situation.
• The Jacobian F (
y (s)) must have full (numerical) rank, which leads to a
lower bound s > smin .
• The local quasi-Gauss-Newton iteration starting at y(s) should converge
sufficiently fast, which leads to an upper bound s ≤ smax .
Prediction strategy. If we use the fact that the Gauss-Newton iteration had
converged on the entering semi-branch towards some point y old , we are led
to the choice
y ∗ − y old s0 := ρ
, ρ<1
t∗1 with some safety factor ρ. With y(s0 ), the Jacobian F (
y (s0 )) and the contraction factors Θ(s0 ) are available in the course of the Gauss-Newton iteration.
Therefore, numerical estimates [smin ], [smax ] can be computed.
In view of (I) above, we require that
ε cond F (
y (s)) < 1
with some prescribed ε > eps, the relative machine precision. Near y ∗ all
determinants (such as dξ , dλ ) are O(s). So we have
γ
cond F (
y (s)) > ,
s
which leads to
s > εγ =: smin .
As γ is unknown, a numerical condition number estimate [cond (·)] is required
to derive the estimate
5
6
[smin ] := ε cond F (
y (s0 )) · s0 .
In view of (II) above, a careful analysis shows that the contraction factor
Θ0 (s) = O(s)
close to y ∗ instead of O(s2 ) in the neighborhood of a regular point y. As a
consequence of the Newton-Kantorovich theorem, we require
5.3 Computation of Simple Bifurcations
Θ0 ≤ Θ =
1
4
279
,
which leads to the stepsize estimate
[smax ] =
Θ
· s0 .
Θ0 (s0 )
In case we had obtained
[smin ] > [smax ]
the computation would have to be terminated suggesting higher precision
arithmetic—thus lowering [smin ].
In the author’s experience, such a situation has never occurred up to now. In
the standard situation
[smin ] [smax ]
some initial steplength s0 can be selected. A typical choice will be
s0 := ρ · [smax ]
for some sufficiently large ρ < 1.
Construction of complete bifurcation diagrams. The implementation
of the whole algorithm—as described in the previous Section 5.2 and the
present Section 5.3—requires careful book-keeping of critical points and of
entering and emanating semi-branches to avoid endless cycling. As an example, before actually iterating towards some conjectured bifurcation point y ∗ ,
the corresponding starting point y 0 should be tested: if it is within the Kantorovich neighborhood of some formerly computed bifurcation point, then
identity of old and new bifurcation point can be assumed; as usual, the test
is based on the local contraction factor criterion Θ0 ≤ 1/4 in agreement
with the sufficient Kantorovich condition (h0 ≤ 1/2). Whenever Θ0 > 1/4,
i.e., when the Kantorovich condition is locally violated, then a possible local
nonuniqueness of a solution is indicated.
Bibliographical Note. The computation of simple bifurcations via the
QR-implementation of Moore’s extended system has been worked out in the
paper [72] by P. Deuflhard, B. Fiedler, and P. Kunkel. The here presented algorithm with quasi-Gauss-Newton method, adaptive stepsize control, computation of turning points and simple bifurcation points has been implemented
in the code ALCON2. An advanced descendant of ALCON2 is the code SYMCON
due to K. Gatermann and A. Hohmann [95] for equivariant parameter dependent nonlinear systems. In this algorithm, symmetries are exploited such that
along each branch symmetry transformations are performed based on Schur’s
lemma. As a consequence, symmetry breaking or symmetry preserving higher
order bifurcations often just show up as simple generic or non-generic bifurcations and can be treated as such. Due to this property the algorithm is
280
5 Parameter Dependent Systems: Continuation Methods
considerably more robust than its predecessor ALCON2. In passing, dynamical
stability of the solutions along each branch can be identified. In order to illustrate the kind of additional results available from SYMCON, we give a rather
challenging illustrative example.
Example 5.4 Hexagonal lattice dome. This well-known challenging equivariant bifurcation problem from continuum mechanics is due to T.J. Healey
[117]. It is known to contain a large number of all kinds of higher order singularities connected with symmetries of the mechanical construction, the most
dominant of which is the symmetry D6 . The problem has been tackled by
SYMCON [95] and is documented in detail in [94]. Figure 5.9 gives parts of the
rather complex total bifurcation diagram associated with two different subsymmetries. As it turned out, the bifurcation diagram computed by SYMCON
revealed hitherto unknown parts.
x
*
*+*
+*
+
*
+* *
*
+
*+
*+
**
*+ +
+
+
** *** *
** *
*+*
*
*
* ** ** *+*
+
+
*+
*+ **+
*
**+
**
x
+
+
+
*
*+
* *
*
*
*
*
** *+ *
* ++ +*+
**
** * ++*++*
* *
+
+
*+
*+
**
*
+
*
+
******
+
+
+
**+
*
+
*+ +
*
*+ *
*
*+
* +* **+ * + * **
+* *
+
*
+*
+ +
*+
* *
+
+* *
*+ **
*
*+
+ +
*
+
*+
*+*+
*
**
+
* *
Fig. 5.9. Hexagonal lattice dome: Bifurcation subdiagrams associated with
partial symmetries. Left: Kleinian group and D6 . Right: D3 and Z24 .
Exercises
Exercise 5.1 Consider the pseudo-arclength continuation method as discussed at the beginning of Section 5.2.1. Study the effect of rescaling of the
parameter
λ −→ σ = λκ .
What kind of continuation method is obtained in the limiting case κ → 0?
Exercise 5.2 Derive feasible stepsize bounds for the classical and the tangent Newton-continuation method using
Exercises
281
a) the affine contravariant Newton-Mysovskikh theorem for the ordinary
Newton method (Theorem 2.12),
b) the affine contravariant Newton-Kantorovich theorem for the simplified
Newton method (Theorem 2.13).
c) On this theoretical basis, design computational estimates for use within
an adaptive stepsize control strategy.
Exercise 5.3 Classical continuation method for nonlinear least squares
problems. For given real parameter λ, let the prediction path be x̂(λ) = x(0).
In the residual based formulation of the Gauss–Newton method as given in
Section 4.2, we write the homotopy for the path x(λ) as
P (x(λ), λ)F (x(λ), λ) ≡ 0 ,
where
P (x, λ) = F (x, λ)F (x, λ)− ,
⊥
P (x, λ) = Im − P (x, λ)
are the corresponding projectors, assumed to be orthogonal.
a) Show that the classical continuation method is of order p = 1.
b) Derive an affine contravariant formula for the feasible stepsize.
c) Design an affine contravariant computational estimate for the order coefficient and consider details for the corresponding adaptive continuation
algorithm.
Exercise 5.4 Tangent continuation method for nonlinear least squares
problems. The notation is the same as in Exercise 5.3. The only new aspect
is that the prediction path now reads
x̂(λ) = x(0) + λẋ(0) .
a) We need an expression for the local path direction ẋ(0). Verify the result
⊥
F (x(0), 0)ẋ(0) + F (x(0), 0)−T F (x(0), 0)[P (x(0), 0)F (x(0), 0), ẋ(0)] =
⊥
− P (x(0), 0)Fλ (x(0), 0) + F (x(0), 0)−T Fλ (x(0), 0)[P (x(0), 0)F (x(0), 0) .
Hint: For the symmetric projector P = AA− in terms of the generalized
(inner) inverse A− apply the formula
⊥
T
⊥
DP = P (DA)A− + P (DA)A−
and, in addition, use special properties at (x(0), 0).
282
5 Parameter Dependent Systems: Continuation Methods
b) Show that under the assumption
⊥
P (x(λ), λ)F (x(λ), λ) ≡ 0
the above equation shrinks to
F (x(0), 0)ẋ(0) + P (x(0), 0)Fλ (x(0), 0) = min ,
which can be satisfied by
ẋ(0) = −F (x(0), 0)− Fλ (x(0), 0) .
c) Discuss the necessary steps to be taken toward an adaptive tangent continuation algorithm.
Part II
DIFFERENTIAL EQUATIONS
6 Stiff ODE Initial Value Problems
This chapter deals with stiff initial value problems for ODEs
ẋ = F (x), x(0) = x0 .
The discretization of such problems is known to involve the solution of nonlinear systems per each discretization step—in one way or the other.
In Section 6.1, the contractivity theory for linear ODEs is revisited in terms
of affine similarity. Based on an affine similar convergence analysis for a simplified Newton method in function space, a nonlinear contractivity theory for
stiff ODE problems is derived in Section 6.2, which is quite different from
the theory given in usual textbooks on the topic. The key idea is to replace
the Picard iteration in function space, known as a tool to show uniqueness in
nonstiff initial value problems, by a simplified Newton iteration in function
space to characterize stiff initial value problems. From this point of view, linearly implicit one-step methods appear as direct realizations of the simplified
Newton iteration in function space. In Section 6.3, exactly the same theoretical characterization is shown to apply also to implicit one-step methods,
which require the solution of a nonlinear system by some finite dimensional
Newton-type method at each discretization step.
Finally, in a deliberately longer Section 6.4, we discuss a class of algorithms
called pseudo-transient continuation algorithms, whereby steady state problems are solved via stiff integration. The latter type of algorithm is particularly useful, when the Jacobian matrix is singular due to hidden dynamical
invariants (such as mass conservation). The affine similar theoretical characterization permits the derivation of an adaptive (pseudo-)time step strategy
and an accuracy matching strategy for a residual based inexact Newton algorithm.
6.1 Affine Similar Linear Contractivity
For the time being, consider a linear ODE system of the kind
ẋ = Ax, x(0) = x0 .
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance
and Adaptive Algorithms, Springer Series in Computational Mathematics 35,
DOI 10.1007/978-3-642-23899-4_6, © Springer-Verlag Berlin Heidelberg 2011
(6.1)
285
286
6 Stiff ODE Initial Value Problems
Formally, system (6.1) can be solved in terms of the matrix exponential
x(t) = exp(At)x0 .
In view of affine similarity as discussed in Section 1.2.2, we start from the
(possibly complex) Jordan decomposition
A = T JT −1 ,
wherein J is the Jordan canonical form consisting of elementary Jordan
blocks for each separate eigenvalue λ(A). Then the (possibly complex) transformation
z := T −1 (x − x̂)
has been shown in Section 1.2.2 to generate an affine similar coordinate frame.
In what follows we will have to work with norms · induced by certain
inner products (·, ·). For simplicity, we may think of the Euclidean norm · induced by the (possibly complex) Euclidean inner product (u, v) = u∗ v with
u∗ the adjoint. If we phrase our subsequent theoretical statements in terms
of the canonical norm
|u| := T −1 u ,
(6.2)
induced by the canonical inner product
u, v = (T −1 u, T −1 v) ,
then such statements will automatically meet the requirement of affine similarity. In this setting, we may define some constant μ = μ(A), allowed to be
positive, zero, or negative, such that
u, Au ≤ μ(A)|u|2 .
(6.3)
This definition is obviously equivalent to
(ū, J ū) ≤ μ(A)ū2 ,
(6.4)
wherein ū = T −1 u. Assuming that the quantity μ is chosen best possible, it
can be shown to satisfy
μ(A) = max
u=0
u, Au
≥ max λi (A) + ,
i
|u|2
≥ 0.
(6.5)
Herein = 0 and equality holds, if the eigenvalue defining μ(A) is simple. It
is an easy task to show that
μ(BAB −1 ) = μ(A)
(6.6)
for any nonsingular matrix B, which confirms that this quantity is indeed
affine similar. In the canonical norm we may obtain the estimate
6.1 Affine Similar Linear Contractivity
287
|x(t)| ≤ exp(μt)|x0 | .
Whenever
μ≤0
(6.7)
holds, then
|x(t)| ≤ |x(0)| ,
which means that the linear dynamical system (6.1) is contractive.
For computational reasons, the Euclidean product (possibly in a scaled variant) is preferred to the canonical product. Suppose that we therefore replace
the above definition (6.3) in terms of the canonical inner product by the
analogous definition
(u, Au)
ν(A) = max
(6.8)
u=0 u2
in terms of the Euclidean product. The thus defined quantity can be expressed
as
ν(A) = λmax 12 (A + AT ) ,
where λmax is the maximum (real) eigenvalue of the symmetric part of the
matrix A. Upon comparison with (6.4) we immediately observe that
μ(A) = ν(J) = λmax 12 (J + J T ) ,
which directly leads to the above result (6.5)—see, e.g., [71, Section 3.2]. From
this we see that the quantities ν(A) and μ(A) may be rather different—in
fact, unless A is symmetric, not even the signs may be the same. Moreover,
in contrast to (6.6), we now have the undesirable property
ν(BAB −1 ) = ν(A) ,
i.e., this quantity is not affine similar. Consequently, contractivity in the
canonical norm | · | does not imply contractivity in the original norm · .
Whenever a relation of the kind
|u| ≤ |v|
is transformed back to the original norm, we can only prove that
u ≤ cond(T )v
in terms of the condition number cond(T ) = T −1 · T ≥ 1, which here
arises as an unavoidable geometric distortion factor. This distortion factor also indicates possible ill-conditioning of the Jordan decomposition as
a whole—which may affect the theoretical presentation in terms of canonical
inner products and norms. Nevertheless, we will stick to a formal notation
in terms of the canonical norm | · | below to make the underlying structure
transparent.
288
6 Stiff ODE Initial Value Problems
6.2 Nonstiff versus Stiff Initial Value Problems
Reliable numerical algorithms are, in one way or the other, appropriate implementations of uniqueness theorems of the underlying analytic problem.
The most popular uniqueness theorem for ODEs is the well-known PicardLindelöf theorem: it is based on the Picard fixed point iteration in function
space (Section 6.2.1) and characterizes the growth of the solution by means
of the Lipschitz constant of the right-hand side—a characterization known
to be appropriate for nonstiff ODEs, but inappropriate for stiff ODEs. As
will be shown in this section, an associated uniqueness theorem for stiff ODEs
can be derived on the basis of a simplified Newton iteration in function space,
wherein the above Lipschitz constant is circumvented by virtue of a one-sided
linear contractivity constant. As a natural spin-off, this theory produces some
common nonlinear contractivity concept both for ODEs (Section 6.2.2) and
for implicit one-step discretizations (see Section 6.3 below).
6.2.1 Picard iteration versus Newton iteration
Consider again the nonlinear initial value problem
ẋ = F (x), x(0) = x0 .
For the subsequent presentation, its equivalent formulation as a Volterra operator equation (of the second kind) is preferable:
τ
G(x, τ ) := x(τ ) − x0 −
F (x(t))dt = 0 .
(6.9)
t=0
This equation defines a homotopy in terms of the interval length τ ≥ 0. Let Γ
denote some neighborhood of the graph of a solution of (6.9). Then Peano’s
existence theorem requires that
L0 := sup F (x) < ∞
Γ
in terms of some pointwise norm · in Rn .
In order to prove uniqueness, the standard approach is to construct the socalled Picard iteration
τ
xi+1 (τ ) = x0 +
F (xi (t))dt
(6.10)
t=0
to be started with x0 (t) ≡ x0 . From this fixed point iteration, one immediately derives
6.2 Nonstiff versus Stiff Initial Value Problems
289
τ
x
i+1
(τ ) − x (τ ) ≤
F (xi (t)) − F (xi−1 (t))dt .
i
t=0
Hence, in order to study contraction, the most natural theoretical characterization is in terms of the Lipschitz constant L1 defined by
F (u) − F (v) ≤ L1 u − v .
With this definition, the sequence {xi } can be shown to converge to some
solution x∗ such that
x∗ (τ ) − x0 ≤ L0 τ ϕ(L1 τ )
with
ϕ(s) :=
(exp(s) − 1)/s
1
s = 0
s = 0.
(6.11)
Moreover, x∗ is unique in Γ . This is the main result of the well-known PicardLindelöf theorem—ignoring for simplicity any distinction between local and
global Lipschitz constants.
A similar term arises in the analysis of one-step discretization methods for
ODE initial value problems. Let p ≥ 1 denote the consistency order of such
a method and τ a selected uniform stepsize assumed to be sufficiently small.
Then the discretization error between the continuous solution x and the discrete solution xτ at some final point T = nτ can be represented in the form
(see, e.g., the ODE textbook by E. Hairer and G. Wanner [114]):
xτ (T ) − x(T ) ≤ Cp · τ p · T · ϕ(L̄1 T ) .
For explicit one-step methods, the coefficient Cp just depends on some bound
in terms of higher derivatives of F . The above discrete Lipschitz constant
L̄1 ≥ L1 is an analog of the continuous Lipschitz constant L1 , this time for
the increment function of the one-step method. In order to assure that the
notion of a consistency order p is meaningful at all, a restriction of the kind
L1 τ ≤ C, C = O(1)
(6.12)
will be needed. Consequently, this characterization is appropriate only for
nonstiff discretization methods.
Historical Note. Originally, it had first been thought that the use of
implicit discretization methods would be the essential item to overcome the
observed difficulties in the numerical integration of what have been called
stiff ODEs—see, for instance, the early fundamental paper by G. Dahlquist
[47]. For implicit one-step methods, the above coefficient Cp is bounded only,
if the discrete solution can be locally continued over each discretization step
290
6 Stiff ODE Initial Value Problems
of length τ . This aspect will be studied in detail in the subsequent Section
6.3. In the next stage of the development of stiff integration, however, it was
recognized that the solution method for the thus arising algebraic equations is
equally important: the early paper of W. Liniger and R.A. Willoughby [145]
pointed out that any fixed point iteration based only on F -evaluations for the
algebraic equations would again bring in restriction (6.12), whereas a Newtonlike iteration could, in principle, avoid the restriction. Much later, so-called
semi-implicit or linearly-implicit discretization methods (such as Rosenbrock
methods, W-methods, or extrapolation methods) were constructed that only
apply one single Newton-like iteration per discretization step. Therefore the
present essence of insight seems to be that nonstiff integration is characterized by sampling of F only, whereas stiff integration requires the additional
sampling of F (x) or an approximation.
With these preparations, a natural approach towards a uniqueness theorem
covering stiff ODEs as well will be to replace the Picard iteration (6.10) by
a Newton iteration. For the ordinary Newton method we would obtain
G (xi )Δxi = −G(xi ),
xi+1 = xi + Δxi
or, in more explicit notation
⎡
τ
F (xi (t))Δxi (t)dt = − ⎣xi (τ ) − x0 −
Δxi (τ ) −
⎤
τ
t=0
F (xi (t))dt⎦ . (6.13)
t=0
However, the above iteration requires global sampling of the Jacobian F (x)
rather than just pointwise sampling as in numerical stiff integration. Therefore, the simplified Newton method will be chosen instead: we just have to
replace
G (xi ) → G (x0 ), x0 (t) ≡ x0
or, equivalently,
F (xi (t)) → F (x0 ) =: A .
The corresponding replacement in (6.13) then leads to
τ
xi+1 (τ ) − A
0
τ
xi+1 (t)dt = x0 +
[F (xi (t)) − Axi (t)]dt .
(6.14)
0
Note that this may be interpreted as a Picard iteration associated with the
formally modified ODE
ẋ − Ax = F (x) − Ax, x(0) = x0 ,
which is the basis for linearly-implicit one-step methods.
6.2 Nonstiff versus Stiff Initial Value Problems
291
6.2.2 Newton-type uniqueness theorems
The above simplified Newton-iteration (6.14) is now exploited with the aim
of proving uniqueness theorems for ODE IVP’s that cover stiff ODEs as well.
In order to guarantee affine similarity, we will define coordinates x ∈ Rn in
such a way that any required (possibly approximate) Jacobian A is already
in its Jordan canonical form J. Consequently, the selected vector norm · is identical to canonical norm as defined in (6.2)—see also the discussion in
Section 6.1. For the time being, we assume that we have an exact initial
Jacobian
F (x0 ) = A = J .
A discussion of the case of an approximate Jacobian will follow subsequently.
Theorem 6.1 In the above notation let F ∈ C 1 (D), D ⊆ Rn . For the Jacobian A := F (x0 ) assume a one-sided Lipschitz condition of the form
(u, Ju) ≤ μu2 ,
where (·, ·) denotes the inner product that induces the norm ·. In this norm,
assume that
F (x0 ) ≤ L0
f or x0 ∈ D
(F (x) − F (x0 ))u ≤ L2 x − x0 u
f or
x, x0 , u ∈ D .
(6.15)
Then, for D sufficiently large, existence and uniqueness of the solution of the
ODE IVP is guaranteed in [0, τ ] such that
τ unbounded , if μτ̄ ≤ −1 ,
τ ≤ τ̄ Ψ (μτ̄ ) , if μτ̄ > −1
with τ̄ := (2L0 L2 )−1/2 and
Ψ (s) :=
ln(1 + s)/s s = 0
1
s = 0.
Proof. Upon performing the variation of constants, we rewrite (6.14) as
τ
i
Δx (τ ) =
d
exp(A(τ − t)) F (xi (t)) − xi (t) dt ,
dt
(6.16)
t=0
where exp(At) denotes the matrix exponential. Within this proof let |·| denote
the standard C 0 -norm:
|u| := max u(t) .
t∈[0,τ ]
In order to study convergence, we set the initial guess x0 (t) ≡ x0 and apply
Theorem 2.5 from Section 2.1.2, which essentially requires that
292
6 Stiff ODE Initial Value Problems
|Δx0 | ≤ α ,
(6.17)
|G (x0 )−1 (G (x) − G (x0 ))| ≤ ω|x − x0 | ,
(6.18)
αω ≤
1
2
.
(6.19)
The rest of the assumptions holds for D sufficiently large. The task is now
to derive upper bounds α, ω and to assure (6.19). With x0 as set above, the
first Newton correction satisfies—compare (6.16)
τ
exp(A(τ − t))F (x0 )dt .
0
Δx (τ ) =
t=0
Hence
τ
Δx (τ ) ≤
exp(As)F (x0 )ds ≤
0
s=0
τ
≤ L0
exp(μs)ds = L0 τ ϕ(μτ ) =: α(τ )
s=0
with ϕ as introduced in (6.11). In order to derive ω(τ ), we introduce the
operator norm in (6.18) by
z := G (y 0 )−1 (G (x0 + w) − G (x0 ))u ,
|z| ≤ ω · |u| · |w| .
Once more by variation of constants, we obtain
τ
exp(A(τ − t))[F (x0 + w) − F (x0 )]udt ,
z(τ ) ≤
t=0
which, similar as above, yields
|z| ≤ L2 · τ · ϕ(μτ ) · |u| · |w| .
Hence, a natural definition is
ω(τ ) := L2 τ ϕ(μτ ) .
Insertion into the Kantorovich condition (6.19) produces
(τ ϕ(μτ ))2 ≤ (2L0 L2 )−1 =: τ̄ 2
or, equivalently,
τ ϕ(μτ ) ≤ τ̄ .
Since μ may have either sign, the main statements of the theorem are an
immediate consequence.
6.2 Nonstiff versus Stiff Initial Value Problems
293
A graphical representation of the above monotone decreasing function Ψ is
given below in Figure 6.3.
Remark 6.1 Upon using the same characterizing quantities μ, τ̄ as above,
an improved theorem has been shown by W. Walter [195] using differential
inequalities—compare his book [194]. Since this theorem is nonconstructive
and does not make a difference for the discretizations to be treated in Section
6.3, it is omitted here (details can be found in the paper [66]).
Nonlinear contractivity. As can be seen in Fig. 6.1 below, the above
theorem (as well as the one in [66]) comes up with a pole at s = −1, which
reflects the condition
μτ̄ ≤ −1
for global boundedness of the solution. This condition may be rewritten as
μ+
2L0 L2 ≤ 0 .
(6.20)
As L2 = 0 in the linear case, the above condition is immediately recognized
as a direct generalization of the linear contractivity condition (6.7). In other
words: the above pole represents global nonlinear contractivity, involving local
contractivity via μ and the part from the nonlinearity in well-separated form.
Ψ(s)
−1
0
Fig. 6.1. Nonlinear contractivity: function Ψ as defined in Theorem 6.1 .
If, instead of the exact Jacobian A = F (x0 ), an approximation error
δA = A − F (x0 )
must be taken into account, then a modification of the above theorem will
be necessary. In view of an affine similar presentation we again assume that
A=J,
which means that the approximate Jacobian is already in Jordan canonical
form and the norm · is identical to the canonical norm.
294
6 Stiff ODE Initial Value Problems
Theorem 6.2 Notation and assumptions as in Theorem 6.1. In addition, let
the Jacobian approximation error be bounded as
δA ≤ δ0 , δ0 ≥ 0 .
Then the results of Theorem 6.1 hold with τ̄ replaced by
τ̂ := τ̄ /(1 + δo τ̄ ) .
Proof. For the proof we apply Theorem 2.6, the convergence theorem for
Newton-like iterations, with G (x0 ) now replaced by M (x0 ), which means
replacing F (x0 ) by A = F (x0 ). With μ now associated with the Jacobian
approximation A , the estimates α(τ ), ω(τ ) carry over. In addition, the assumptions (6.17) up to (6.19) must be extended by
|M (x0 )−1 (G (x0 ) − M (x0 ))| ≤ δ̄0 < 1 .
Upon defining
(6.21)
z := M (x0 )−1 (G (x0 ) − M (x0 ))u
a similar estimate as in the proof of Theorem 6.1 leads to
τ
z(τ ) ≤
exp(A(τ − t)) · δA · udt ≤ δ0 τ ϕ(μτ )|u| .
t=0
Hence, the above condition (6.21) shows up with the specification
δ̄0 := δ0 τ ϕ(μτ ) .
Insertion into the modified Kantorovich condition (2.23)
αω
≤
(1 − δ̄0 )2
1
2
then yields
τ ϕ(μτ ) ≤ τ̄ /(1 + δ0 τ̄ ) =: τ̂ .
Note that condition (6.21) is automatically satisfied, since
δ̄0 = δ0 τ ϕ(μτ ) ≤ δ0 τ̄ /(1 + δ0 τ̄ ) < 1 ,
which completes the proof.
Finally, we want to emphasize that all above results also hold, if the norm ·
is not identified with the canonical norm |·|, but allowed to be a general vector
norm. However, as already worked out at the end of Section 6.1, this would
include a tacit deterioration of all results, since then the one-sided Lipschitz
6.3 Uniqueness Theorems for Implicit One-step Methods
295
constant μ may be rather off scale, if not even nonnegative—compare again
the definitions (6.3) and (6.8).
Remark 6.2 The experienced reader will be interested to know whether
these results carry over to differential-algebraic equations (DAEs) as well.
Unfortunately, this causes some difficulty, which can already be seen in the
simple separable DAE of the form
y = f (y, z) ,
0 = g(y, z) .
In this case, the differential part f and the variable y suggest affine similarity, whereas the equality constrained part g would require some affine
covariance or contravariance. For this reason, a common affine invariant theoretical framework is hard to get, if at all possible. Up to now, more subtle
estimation techniques use a characterization in terms of perturbation parameters , which by construction do not allow for any affine invariance concept.
6.3 Uniqueness Theorems for Implicit One-step
Methods
A natural requirement for any discretization π of the above ODE initial value
problem will be that it passes basic symmetries of the underlying continuous
problem on to the generated discrete problem. In particular, we will require
that the diagram in Figure 6.2 commutes, which ensures that yπ = Bxπ
holds whenever y = Bx. Among the discretization methods satisfying this
requirement, we will restrict our attention to implicit and linearly implicit
one-step methods, also called Runge-Kutta methods.
B
x
y
B −1
π
π
?
xπ
B
-
?
yπ
B −1
Fig. 6.2. Affine invariance under discretization π
296
6 Stiff ODE Initial Value Problems
As an extension of Section 6.1, any affine similar discretization of linear
ODEs can also be treated in affine similar terms. This idea directly leads
to G. Dahlquist’s linear scalar model equation [47]
ẋ = λx,
x(0) = 1 .
Therefore, linear contractivity of implicit discretizations as well as of linearly
implicit discretizations can be treated just as described in usual numerical
ODE textbooks—see, e.g., [114, 115].
Things are different with respect to nonlinear contractivity. First, recall from
Section 6.2.1 that the simplified Newton iteration (6.14) for the continuous
ODE problem may also be regarded as a Picard iteration (6.10) for the ODE
ẋ − Ax = F (x) − Ax .
This ODE is the starting point for linearly implicit one-step discretizations
(such as Rosenbrock methods, W-methods, or extrapolation methods), which
just discretize the above left hand side implicitly and the above right hand
side explicitly. Therefore, linearly implicit discretizations may be interpreted
as direct realizations of the simplified Newton iteration in function space. Of
course, they should also observe the local timestep restrictions as worked
out for the continuous problem in Section 6.2.2. For the special case of the
linearly implicit Euler discretization we refer to the residual analysis given in
the subsequent Section 6.4.
Here we concentrate on implicit one-step discretizations. In such discretizations the discrete system comprises a nonlinear algebraic system, which again
brings up the question of local continuation. We will be interested to see, in
which way some kind of nonlinear contractivity is inherited from the continuous initial value problem to various implicit one-step methods. In order to
permit a comparison with Section 6.2.2, we will again assume that the coordinates have been chosen such that the local Jacobian matrix A ≈ F (x0 ) is
already in Jordan canonical form—which implies that the canonical product
and norm are identical to the Euclidean product and norm. To start with,
we exemplify the formalism at a few simple cases.
Implicit Euler discretization. In each step of this discretization, we must
solve the n algebraic equations
G(x, τ) := x − x0 − τ F (x) = 0 ,
(6.22)
which represent a homotopy in Rn with embedding in terms of the stepsize
τ —say τ ≥ 0. The Newton-like iteration for solving this system is
(I − τ A)Δxi = −(xi − x0 − τ F (xi )),
where δA := A − F (x0 ) = 0 will be assumed.
xi+1 = xi + Δxi ,
(6.23)
6.3 Uniqueness Theorems for Implicit One-step Methods
297
Theorem 6.3 Assumptions and notation as in Theorems 6.1 and 6.2 above.
Then the Newton-like iteration (6.23) for the implicit Euler discretization
converges to a unique solution for all stepsizes
τ unbounded , if μτ̂ ≤ −1 ,
τ ≤ τ̂ ΨD (μτ̂ ) , if μτ̂ > −1 ,
where
ΨD (s) := (1 + s)−1 .
Proof. Once more, Theorem 2.6 is applied, here to the finite-dimensional
homotopy (6.22). The Jacobian approximation A ≈ F (x0 ) leads to the approximation
I − τ A =: M (x0 ) ≈ F (x0 ) ,
which is used in the definition of the affine covariant Lipschitz constant
M (x0 )−1 (F (u) − F (v)) ≤ ω(τ )u − v ,
the first correction bound
Δx0 = M (x0 )−1 F (x0 ) ≤ α(τ )
and the approximation measure
M (x0 )−1 (M (x0 ) − F (x0 )) ≤ δ̄0 (τ ) < 1 .
With these definitions, the modified Kantorovich condition here reads
αω
≤ 12 .
(6.24)
(1 − δ̄o )2
Upon using similar techniques as in the proof of Theorem 6.2 above, we come
up with the estimates:
α(τ ) := τ L0 /(1 − μτ ),
ω(τ ) := hL2 /(1 − μτ ),
δ̄0 (τ ) := τ δ0 /(1 − μτ ),
where the quantities L0 , L2 , δ0 are the same as in Section 6.2.2. Insertion into
condition (6.24) then yields, for μτ < 1:
τ
≤ τ̂
1 − μτ
or, equivalently,
τ ≤ τ̂ /(1 + μτ̂ ) .
This is the main statement of the theorem. Finally, note that for μ > 0
μτ ≤ μτ̂ /(1 + μτ̂ ) < 1 ,
which assures the above requirement. The case μ ≤ 0 is trivial.
The intriguing similarity of Theorems 6.2 and 6.3 is illustrated in Figure 6.3.
298
6 Stiff ODE Initial Value Problems
ΨD (s)
Ψ(s)
−1
0
Fig. 6.3. Nonlinear contractivity inherited: function Ψ (continuous case)
versus function ΨD (discrete case).
Implicit trapezoidal rule. This discretization requires the solution of the
n in general nonlinear equations
G(x) := x − x0 − 12 τ (F (x) + F (x0 )) = 0 .
Standard Newton-like iteration leads to the steplength restriction
√
a) τ unbounded, if μτ̄ ≤ − 2 ,
√
μτ̄
b) τ ≤ τ̄ 2ΨD √
.
2
(6.25)
(6.26)
Observe that the pole of Ψ at s = −1 is not preserved here, so that nonlinear
contractivity is not correctly inherited from the continuous case.
Implicit midpoint rule. This discretization leads to the n equations
x + x0
G(x) := x − x0 − τ F
= 0,
(6.27)
2
which, along similar lines of derivation, yields the stepsize bounds
a)
τ unbounded, if μτ̄ ≤ −1 ,
b)
τ ≤ 2τ̄ ΨD (μτ̄ ) .
(6.28)
Here the pole is correctly preserved. Moreover, less restrictive bounds appear.
Summarizing, the implicit trapezoidal rule and the implicit midpoint rule
have the same linear contractivity properties, but different nonlinear contractivity properties. From the nonlinear contractivity point of view, the implicit
midpoint rule is clearly preferable. Both proofs are just along the lines of
the proof for the implicit Euler method and therefore left as Exercise 6.1.
Of course, one would really like to characterize the whole subclass of those
one-step methods that preserve the pole exactly—a question left to future
research.
6.4 Pseudo-transient Continuation for Steady State Problems
299
6.4 Pseudo-transient Continuation for Steady State
Problems
In this section we consider the case that the solution of a nonlinear system
F (x) = 0 can be interpreted as steady state of a dynamical system of the kind
ẋ = F (x) .
(6.29)
Already from mere geometrical insight it is clear that such an approach will
only work, if the fixed point of the dynamical system is attractive in a sufficiently large neighborhood. As an example, stiff integration towards a hyperbolic fixed point might come close to the fixed point for a while and run
away afterwards. Exceptions will be possible only for a measure zero set of
starting points x0 .
Dynamical invariants. This type of invariants occurs rather frequently in
dynamical systems causing singular Jacobian matrices F (x) for all arguments
x—which prohibits the application of standard Newton methods.
Example: mass conservation. Suppose the above ODE (6.29) describes some
reaction kinetic model. Then mass conservation shows up as
eT x(t) = eT x0 ,
where eT = (1, . . . , 1). This implies
eT ẋ = eT F (x) ≡ 0,
x ∈ D ⊂ Rn ,
F (x) = 0 .
By differentiation with respect to x we obtain
eT F (x)F (x) ≡ 0,
F (x) = 0
and hence every Jacobian has a zero eigenvalue with left eigenvector e. If we
define the orthogonal projectors
P ⊥ :=
1 T
ee ,
n
P = I − P⊥ ,
then we can write equivalently
P ⊥ F (x) = 0 .
(6.30)
Of course, naive application of any standard Newton method would fail in
this situation. In this special case, a modification is possible that makes the
Newton methods nevertheless work—see, e.g., Exercise 6.3.
In the general case, however, more than one dynamical invariant exists, most
of them unspecified or even unknown, so that (6.30) holds again, now for an
unknown projector P such that
P ⊥ ẋ = P ⊥ F (x) = 0
=⇒
P ⊥ F (x) = 0 .
(6.31)
Clearly, Newton methods cannot be modified to work without full knowledge
about all dynamical invariants and are therefore bound to fail.
300
6 Stiff ODE Initial Value Problems
Fixed point iterations. In contrast to the other affine invariance classes
of nonlinear problems, affine similarity also holds for fixed point iterations
Δxk = xk+1 − xk = αF (xk )
with a parameter α to be adapted. From equation (6.31) we see that such an
iteration automatically realizes
P ⊥ Δx = 0 .
Pseudo-transient continuation. The same property can be shown to hold
for any linear combination of Newton and fixed point iteration. A popular
technique is the so-called pseudo-transient continuation method
(I − τ A) Δx = F (x0 ),
x(τ ) = x0 + τ Δx
(6.32)
with timestep τ to be adapted and A = F (x0 ) or a Jacobian approximation.
The above iteration is just a special stiff discretization of the ODE (6.29),
known as the linearly implicit Euler discretization.
Of course, in order to obtain the solution, we may directly solve the time
dependent system (6.29) by any numerical stiff integrator up to the steady
state. In what follows, however, we want to restrict our attention to the simple
case of the linearly implicit Euler discretization.
6.4.1 Exact pseudo-transient continuation
We now want to study an iterative method for the numerical solution of the
nonlinear System F (x) = 0 based on the linearly implicit Euler discretization
(6.32). Throughout this section we will assume that we can evaluate an exact
Jacobian A = F (x) and solve the linear system (6.32) by direct elimination
techniques.
As worked out in detail in Section 6.1, the problem itself is invariant under
affine similarity transformation, which would suggest some theoretical treatment in terms of canonical norms and inner products. Usual stiff integration
focuses on the accuracy of the solution which naturally belongs to an affine
covariant setting. For reasons of numerical realization, however, we need to
study the convergence of the iteration in terms of its residual behavior—
which leads to an affine contravariant setting. For that reason, we will need
to replace the canonical norm | · | (see Section 6.1) by some Euclidean norm
·, possibly scaled. Accordingly (·, ·) will denote the Euclidean inner product,
also possibly scaled.
Let x(τ ) denote the homotopy path defined by (6.32) and starting at the
point x(0) = x0 . Before we actually study the residual norm F (x(τ )), the
following auxiliary result will be helpful.
6.4 Pseudo-transient Continuation for Steady State Problems
301
Lemma 6.4 Notation as just introduced with A ≈ F (x0 ). Then the residual
along the homotopy path x(τ ) starting at x0 satisfies
τ
F (x(τ )) = (I−τ A)
−1
(F (x(σ)) − A) (I−σA)−2 F (x0 )dσ . (6.33)
F (x0 )+
σ=0
Proof. Taylor’s expansion of the residual yields
τ
F (x(τ ))
F (x(σ))ẋ(σ)dσ
= F (x0 ) +
σ=0
τ
(F (x(σ)) − A) ẋ(σ)dσ .
= F (x0 ) + A(x(τ ) − x0 ) +
σ=0
Upon differentiating the homotopy (6.32) with respect to τ , we obtain
(I − τ A)ẋ = F (x0 ) + A(x(τ ) − x0 ) = F (x0 ) + τ A(I − τ A)−1 F (x0 )
and therefore
ẋ(τ ) = (I − τ A)−2 F (x0 ) ,
which then readily leads to the result of the lemma.
Discussion of Lipschitz conditions. With the above representation at
hand, the question is now how to formulate first and second order Lipschitz
conditions in view of theoretical estimates. The switch from the canonical
norms in Sections 6.2 and 6.3 to the Euclidean norm here implies changes
in all our definitions of first and second order Lipschitz constants below.
Needless to mention that we are bound to lose the nice property of affine
similarity in all our characterizing quantities. Instead all of our estimates will
now depend on the scaling of the residual (to be carefully handled).
First order Lipschitz condition: linear contractivity. We may employ (6.8)
to define some one-sided Lipschitz constant ν. Recall, however, that due to
dynamical invariants, zero eigenvalues will occur in the Jacobian, which implies that ν ≥ 0—just apply the definition (6.8) again. Therefore, in order to
take care of dynamical invariants, we will restrict our attention to iterative
corrections in the subspace
SP = {u ∈ Rn | P ⊥ u = 0} .
Then the inequality
(u, Au) ≤ νu2 ,
u ∈ SP
is equivalent to
(P u, (P AP )P u) ≤ νP u2 .
302
6 Stiff ODE Initial Value Problems
With this modified definition, the case ν < 0 may well happen even in the
presence of dynamical invariants.
Since Δx(τ ) ∈ SP , we may insert it into the above definition and obtain
ν̂(τ ) =
(Δx, AΔx)
≤ν.
Δx2
(6.34)
If we multiply equation (6.32) by Δx from the left, we obtain
Δx2
= τ (Δx, AΔx) + (Δx, F (x0 ))
= τ ν̂(τ )Δx2 + (Δx, F (x0 ))
≤ τ ν̂(τ )Δx2 + ΔxF (x0 )
≤ τ νΔx2 + ΔxF (x0 ) ,
which then leads to the estimates
Δx ≤
F (x0 )
F (x0 )
≤
.
1 − ν̂τ
1 − ντ
(6.35)
Moreover, since
Δx(τ ) = F (x0 ) + O(τ ) ,
we also have
ν̂(0) =
(F (x0 ), AF (x0 ))
≤ν.
F (x0 )2
(6.36)
This quantity can be monitored even before the linear equation (6.32) is
actually solved. It plays a key role in the residual reduction process, as shown
in the following lemma.
Lemma 6.5 Let ν̂(0) < 0 as defined in (6.36). Then there exists some τ ∗ > 0
such that
F (x(τ )) < F (x0 )
and
ν̂(τ ) < 0
for all
τ ∈ [0, τ ∗ [ .
Proof. By differentiating the residual norm with respect to τ we obtain
d
F (x(τ ))2 |τ =0 = 2(F (·)T F (·), ẋ(τ ))|τ =0
dτ
= 2(F (x0 ), AF (x0 )) = 2ν̂(0)F (x0 )2 < 0 .
Since both F (x(τ )) and the norm · are continuously differentiable, there
exists some nonvoid interval w.r.t. τ , wherein the residual norm decreases—
compare the previous Lemma 3.2. The proof of the statement for ν̂(τ ) uses
the same kind of argument.
6.4 Pseudo-transient Continuation for Steady State Problems
303
In other words: if at the given starting point x0 the condition ν̂(0) < 0 is not
satisfied, then the pseudo-transient continuation method based on residual
reduction cannot be expected to work at all. Recall, however, the discussion
at the end of Section 6.1 which pointed out that residual reduction is not
coupled to canonical norm reduction.
Second order Lipschitz condition. Here we may start from the affine similar
Lipschitz condition (6.15) and replace the canonical norm | · | therein by the
Euclidean norm · . Thus we take the fact into account that in the affine
similar setting domain and image space transform in the same way.
Convergence analysis. With these preparations we are now ready to state
our main result.
Theorem 6.6 Notation as in the preceding Lemma 6.4, but with A = F (x0 )
and partly L0 = F (x0 ). Let dynamical invariants show up via the properties
F (x) ∈ SP . Assume the one-sided first order Lipschitz condition
(u, Au) ≤ νu2
f or
u ∈ SP ,
ν<0
and the second order Lipschitz condition
F (x) − F (x0 ) u ≤ L2 x − x0 u .
Then the following estimate holds
1
L0 L2 τ 2 F (x0 )
F (x(τ )) ≤ 1 + 2
.
1 − ντ
1 − ντ
(6.37)
(6.38)
From this, residual monotonicity
F (x(τ )) ≤ F (x0 )
is guaranteed for all τ ≥ 0 satisfying the sufficient condition
ν + ( 12 L0 L2 − ν 2 )τ ≤ 0 .
(6.39)
L 0 L2 > ν 2 ,
(6.40)
Moreover, if
then the theoretically optimal pseudo-timestep is
τopt =
leading to a residual reduction
F (x(τ )) ≤ 1 −
|ν|
L 0 L2 − ν 2
1 2
2ν
L 0 L2
F (x0 ) < F (x0 ) .
(6.41)
304
6 Stiff ODE Initial Value Problems
Proof. We return to the preceding Lemma 6.4. Obviously, the first and the
second right hand terms in equation (6.33) are independent. Upon recalling
(6.35) for the first term, we immediately recognize that, in order to be able
to prove residual reduction, we necessarily need the condition ν < 0, which
means ν = −|ν| throughout the proof. For the second term we may estimate,
again recalling (6.35),
τ
(F (x(σ)) − F (x0 )) (I − σA)−2 F (x0 )dσ
σ=0
τ
≤ L2
σ=0
τ
≤ L2
σ=0
x(σ) − x0 (I − σA)−2 F (x0 )dσ
σF (x0 )2
dσ
(1 − σν)3
= 12 L2 F (x0 )2 τ 2 (1 − ντ )−2 .
Combination of the two estimates then directly confirms (6.38), which we
here write as
F (x(τ )) ≤ α(τ )F (x0 ) ,
in terms of
α(τ ) = 1 − ντ + 12 L0 L2 τ 2 /(1 − ντ )2 .
Upon requiring α(τ ) ≤ 1, we obtain the equivalent sufficient condition (6.39).
Finally, in order to find the optimal residual reduction, a short calculation
shows that
L 0 L2 τ
α̇(τ ) = ν +
/(1 − ντ )2 .
1 − ντ
An interior minimum can arise only for α̇(τ ) = 0, which is equivalent to (6.41)
under the condition (6.40). Insertion of τopt into the expression for α(τ ) then
completes the proof.
From the above condition (6.39) we may conclude: if
ν+
1
2
2L0 L2 ≤ 0 ,
then τ is unbounded for local continuation. Obviously, this is the residual
oriented nonlinear contractivity condition to be compared with the error oriented relation (6.20). (The difference in the prefactor just indicates that there
we needed to show uniqueness in addition.) If
ν+
1
2
2L0 L2 > 0 ,
(6.42)
6.4 Pseudo-transient Continuation for Steady State Problems
305
then the pseudo-timestep is bounded according to
τ≤
|ν|
.
− |ν|2
1
L L
2 0 2
Note that condition (6.40) is less restrictive than (6.42) so that either unbounded or bounded optimal timesteps may occur.
Adaptive (pseudo-)timestep strategy. In the spirit of the whole book
we now want to derive an adaptive strategy based on the theoretical optimal
pseudo-timestep (6.41), which we repeat for convenience
τopt =
|ν|
.
L0 L 2 − ν 2
The above expression can be rewritten in implicit form as
τopt =
|ν|(1 − ντopt )
.
L0 L2
(6.43)
In passing we note that from this representation we roughly obtain
τopt ∼
1
1
=
,
L0
F (x0 )
which gives some justification for a quite popular heuristic strategy: new
timesteps are proposed on the basis of successful old ones via
τnew =
F (xold )
τold .
F (xnew )
(6.44)
For reference see, e.g., the recent paper [133] by C.T. Kelley and D.E. Keyes,
where also a whole class of further heuristics is mentioned. A different approach is taken by S.B. Hazra, V. Schulz, J. Brezillon, and N. Gauger in [116]
where in a fluid dynamical problem no overall timestep exists; this approach
is not treated here.
Here, however, we want to exploit the structure of (6.43) in a different way
by rewriting it in the form
Δx(τopt )L2 τopt ≤
L0 L2
= |ν| .
1 − ντopt
On this basis, we replace τopt by the upper bound
τ̄opt =
|ν|
≥ τopt .
L2 Δx(τ )
So we are left with the task of identifying cheap computational estimates
[ν] ≤ ν < 0, [L2 ] ≤ L2 to replace the unknown theoretical quantities ν, L2 .
Once this is achieved, we can compute the corresponding pseudo-timestep
306
6 Stiff ODE Initial Value Problems
[τopt ] =
|[ν]|
≥ τ̄opt ≥ τopt .
[L2 ]Δx(τ )
(6.45)
As for the estimation of ν, we may exploit (6.34) in a double way: First,
whenever
Δx(τ ) ≥ F (x0 ) ,
then we know that ν ≥ [ν] ≥ 0 is guaranteed, which means that we should
terminate the iteration. Second, we may recognize that
[ν]τ = ν̂(τ )τ = τ
(Δx, AΔx)
(Δx, Δx − F (x0 ))
=
≤ ντ
2
Δx
Δx2
(6.46)
gives us a quite cheap estimation formula for ν. As for the estimation of L2 ,
we may rearrange terms in the proof of Theorem 6.6 to obtain
τ
F (x(τ )) − Δx(τ )
(F (x(σ)) − F (x0 )) (I − σA)−2 F (x0 )dσ
≤
σ=0
τ
≤ L2
x(σ) − x0 (I − σA)−1 Δx(σ)dσ .
σ=0
If we approximate the integral by the trapezoidal rule, we arrive at
F (x(τ )) − Δx(τ )
≤
1
2 2
2 L2 Δx(τ ) τ /(1
≤
1
L Δx(τ )2 τ 2
2 2
− ντ ) + O(τ 4 )
+ O(τ 4 ) .
Note that already the approximation term, ignoring the O(τ 4 ) term, gives
rise to the upper bound
F (x(τ )) − Δx(τ ) ≤ 12 L2 F (x0 )2 τ 2 /(1 − ντ )2 ,
which is the basis of the derivation of Theorem 6.6. Hence, we may well regard
[L2 ] =
2F (x(τ )) − Δx
≤ L2 + O(τ 2 )
τ 2 Δx2
as a suitable computational estimate for L2 . Upon collecting all above estimates and inserting them into (6.45), we arrive at the following pseudotimestep suggestion
[τopt ] =
|(Δx(τ ), F (x0 ) − Δx(τ ))|
τ.
2Δx(τ )F (x(τ )) − Δx(τ )
On this basis, an adaptive τ -strategy can be realized in the usual two modes,
a correction and a prediction strategy: If in the iterative step x0 −→ x(τ ) the
residual norm does not decrease, then the actual step size τ is replaced by
6.4 Pseudo-transient Continuation for Steady State Problems
307
[τopt ] < τ ; if the residual norm decreases successfully, then the next step is
started with the trial value [τopt ]. Finally, note that the above strategy will
just terminate, if the steady state to be computed is not attractive in the
(residual) sense that [ν] ≥ 0. For [ν] → 0− , the suggested stepsize behaves
like [τopt ] → 0+ - as to be reasonably required.
6.4.2 Inexact pseudo-transient continuation
Suppose the linear system (6.32) is so large that we cannot but solve it
iteratively (i = 0, 1, . . .):
(I − τ A)δxi = F (x0 ) − ri ,
xi (τ ) = x0 + τ δxi .
(6.47)
Herein ri represents the iterative linear residual, δxi the corresponding inexact correction, and xi (τ ) the approximate homotopy path instead of the exact
x(τ ). To start the iteration, let x0 (τ ) = x0 so that δx0 = 0 and r0 = F (x0 ).
If we want to minimize the residuals within each iterative step, we are directly led to GMRES—see Section 1.4 and the notation therein. In terms of the
Euclidean norm · we define the approximation quantities
ηi :=
ri < 1 f or
F (x0 )
i = 1, 2, . . . .
Recall that GMRES assures ηi+1 ≤ ηi , in the generic case even ηi+1 < ηi .
Moreover, due to the residual minimization property and r0 = F (x0 ), we
have
F (x0 ) − ri 2 = (1 − ηi2 )F (x0 )2 .
In the present context of pseudo-transient continuation, we may additionally
observe that GMRES realizes the special structure
δxi (τ ) = Vi zi (τ )
and Hi (τ ) = (Ii , 0)T + τ Ĥi .
Herein Vi is just the orthonormal basis of the Krylov space Ki (r0 , A) and
Ĥi is a Hessenberg matrix like Hi (τ ), but also independent of τ . On this
basis, we see that dynamical invariants are correctly treated throughout the
iteration. The proof of these properties is left as Exercise 6.5. The special
structure permits computational savings when the same system is solved for
different pseudo-timesteps τ .
Convergence analysis. As before, we first analyze the convergence behavior theoretically as a basis for the subsequent derivation of an adaptive
algorithm, which here will have to include the matching of inner and outer
iteration. For this purpose we need to modify Lemma 6.4.
308
6 Stiff ODE Initial Value Problems
Lemma 6.7 Notation as in Lemma 6.4 with A ≈ F (x0 ). Then the residual
along the approximate homotopy path xi (τ ) starting at x0 satisfies
F (x(τ )) − ri = (I − τ A)−1 (F (x0 ) − ri )
τ
(F (xi (σ)) − A) (I − σA)−2 (F (x0 ) − ri ) dσ .
+
σ=0
Proof. The proof is just an elementary modification of the proof of Lemma
6.4. For example, if we differentiate the homotopy (6.47) with respect to τ ,
we now obtain
ẋi (τ ) = (I − τ A)−2 (F (x0 ) − ri ) .
Further details can be omitted.
Theorem 6.8 Notation as in the preceding Lemma 6.7. Let A = F (x0 ) and
partly L̃0 = 1 − ηi2 F (x0 ). Assume that dynamical invariants show up via
the properties F (x) ∈ SP . Then, with the Lipschitz conditions
(u, Au) ≤ νu2 ,
and
ν < 0,
f or
u ∈ SP
F (x) − F (x0 ) u ≤ L2 x − x0 u ,
the estimates
F (x(τ )) − ri ≤
and
F (x(τ )) ≤
1+
1
2
2 L̃0 L2 τ
1 − ντ
1 − ηi2
ηi +
1 − ντ
1+
F (x0 ) − ri 1 − ντ
1
L̃ L τ 2
2 0 2
1 − ντ
F (x0 )
hold. Let
9
s(ηi ) :=
1 − ηi
>
1 + ηi
1
2
or, equivalently,
ηi <
3
5
.
(6.48)
is guaranteed for all τ ≥ 0 satisfying the sufficient condition
1 − s(ηi ) + (2s(ηi ) − 1)ντ + 12 L̃0 L2 − s(ηi )ν 2 τ 2 ≤ 0 .
(6.49)
Then residual monotonicity
F (x(τ )) ≤ F (x0 )
Assume further that
6.4 Pseudo-transient Continuation for Steady State Problems
4
L L
5 0 2
> ν2
and that GMRES has been continued until
ηi + 1 − ηi2 < 1 +
309
(6.50)
1 2
ν
2
L0 L2
.
(6.51)
Then the theoretically optimal pseudo-timestep is
τopt =
|ν|
L̃0 L2 − ν 2
(6.52)
leading to the estimate
F (x(τ )) − ri ≤
1−
1 2
2ν
L̃0 L2
and to the residual reduction
F (x(τ )) ≤ ηi + 1 − ηi2 −
F (x0 ) − ri < F (x0 ) − ri 1 2
ν
2
L0 L 2
F (x0 ) < F (x0 ) .
(6.53)
Proof. We return to the preceding Lemma 6.7 and modify the proof of
Theorem 6.6 carefully step by step. For example, the second order term may
be estimated as
τ
(F (x(σ)) − F (x0 )) (I − σA)−2 (F (x0 ) − ri )dσ
σ=0
≤ 12 L2 F (x0 ) − ri 2 τ 2 (1 − ντ )−2 .
Combination of estimates then directly confirms
F (xi (τ )) − ri ≤ ᾱi (τ )F (x0 ) − ri in terms of
ᾱi (τ ) = 1 − ντ + 12 L̃0 L2 τ 2 /(1 − ντ )2 ,
from which we obtain
F (xi (τ )) ≤ αi (τ )F (x0 )
with
αi (τ ) = ηi +
1 − ηi2 ᾱi (τ ) .
Upon requiring α(τ ) ≤ 1, we have
ηi + 1 − ηi2 ᾱi (τ ) ≤ 1 ,
310
6 Stiff ODE Initial Value Problems
which is equivalent to
ᾱi (τ ) ≤ s(ηi ) ≤ 1 .
(6.54)
From this, we immediately verify the sufficient condition (6.49). Note that
2s − 1 > 0, which is just condition (6.48), is necessary to have at least one
negative term in the left hand side of (6.49).
Finally, in order to find the optimal residual reduction, a short calculation
shows that
2
1
−
η
L̃
L
τ
0
2
i
˙ )=
α̇(τ ) = 1 − ηi2 ᾱ(τ
ν+
.
(1 − ντ )2
1 − ντ
˙ ) = 0, which is equivalent to (6.52)
For the interior minimum we require ᾱ(τ
under the condition (6.50), where
1 − ηi2 ≥ 1 − ( 35 )2 = 45
has been used. Insertion of τopt into the expression for α(τ ) then leads to
1 2 ν
F (xi (τ )) − ri ≤ 1 − 2
F (x0 ) − ri L̃0 L2
and eventually to (6.53). In order to assure an actual residual reduction, condition (6.54) must also hold for τopt , which confirms the necessary condition
2
(6.51). Note
√that the scalar function ηi + 31 − ηi is monotonically increasing
1
for ηi < 2 2 ≈ 0.7, hence also for ηi < 5 = 0.6. Therefore GMRES may be
just continued until the relation (6.51) is satisfied. This completes the proof.
Adaptive (pseudo-)timestep strategy. We follow the line of the derivation for the exact pseudo-transient continuation in Section 6.4.1. For convenience, we repeat the expression
τopt =
|ν|
,
L̃0 L2 − ν 2
which can be rewritten in implicit form as
τopt =
|ν|(1 − ντopt )
.
L̃0 L2
Recall now that
δxi (τ ) = (I − τ A)−1 (F (x0 ) − ri ) ≤
which directly implies
F (x0 ) − ri L̃0
=
,
1 − ντ
1 − ντ
(6.55)
6.4 Pseudo-transient Continuation for Steady State Problems
τ̄opt =
311
|ν|
≥ τopt .
L2 δxi (τ )
So we need to compute the pseudo-timestep
[τopt ] =
|[ν]|
≥ τ̄opt ≥ τopt
[L2 ]δxi (τ )
(6.56)
in terms of the appropriate estimates of the unknown theoretical quantities
ν, L2 .
As for the estimation of ν, we exploit (6.55). Whenever
δxi (τ ) ≥ F (x0 ) − ri ,
then we know that ν ≥ 0 is guaranteed and the iteration must be terminated.
Moreover, the relation
[ν]τ = τ
(δxi , Aδxi )
(δxi , δxi − F (x0 ) + ri )
=
≤ ντ
δxi 2
δxi 2
supplies an estimation formula for ν. As for the estimation of L2 , we revisit
Lemma 6.7 to obtain
F (xi (τ )) − ri − δxi (τ )
τ
(F (xi (σ)) − F (x0 )) (I − σA)−2 (F (x0 ) − ri ) dσ
≤
σ=0
τ
≤ L2
σ=0
≤
1
2
2 L2 τ
xi (σ) − x0 (I − σA)−1 δxi (σ)dσ
L̃20
.
(1 − ντ )2
If we approximate the above integral by the trapezoidal rule (before using the
final estimate), we arrive at
F (xi (τ )) − ri − δxi (τ )
≤
1
L δxi (τ )2 τ 2 /(1
2 2
≤
1
2
2
2 L2 τ δxi (τ )
− ντ ) + O(τ 4 )
+ O(τ 4 ) .
Already the first right hand term gives rise to the above upper bound—
compare (6.55). Hence, as in Section 6.4.1, we will pick
[L2 ] =
2F (xi (τ )) − ri − δxi (τ )
≤ L2 + O(τ 2 )
τ 2 δxi (τ )2
as computational estimate for L2 . Upon inserting the two derived estimates
into (6.56), we arrive at the pseudo-timestep estimate
312
6 Stiff ODE Initial Value Problems
[τopt ] =
|(δxi (τ ), F (x0 ) − ri − δxi (τ ))|
τ.
2δxi (τ )F (xi (τ )) − ri − δxi (τ )
On this basis, an adaptive τ -strategy can again be realized as in the case of
the exact pseudo-transient continuation method.
Finally, we want to mention that the iterative version of the pseudo-transient
continuation method still works in the case of unbounded timestep. To see
this, just rewrite (6.47) in the form
1
I − A (xi (τ ) − x0 ) = F (x0 ) − ri .
τ
Herein τ → ∞ is possible leaving xi (τ ) − x0 well-defined even in the presence
of singular Jacobian A caused by dynamical invariants: This is due to the fact
that GMRES (like any Krylov solver) keeps the nullspace components of the
solution unchanged, so that P ⊥ (xi (∞) − x0 ) = 0 is guaranteed throughout
the iteration.
Preconditioning. If we multiply the nonlinear system by means of some
nonsingular matrix M from the left as
M ẋ = M F (x) = 0 ,
then GMRES will have to work on the preconditioned residuals M ri and adaptivity must be based on norms M · . Note that it is totally unclear,
whether such a transformation leads to the necessary linear contractivity
result ν(M A) < 0 for the preconditioned system with A ≈ F (x0 ).
Preconditioning from the right will just influence the convergence speed of
GMRES without changing the above derived adaptivity devices.
Matrix-free realization. Sometimes the inexact pseudo-continuation method is realized in a matrix-free variant using the first order approximation
Aδx ≈ F (x + δx) − F (x) .
A numerically stable realization will use automatic differentiation as suggested by A. Griewank [112].
Exercises
Exercise 6.1 Prove the results (6.26) for the implicit trapezoidal rule (6.25)
and (6.28) for the implicit midpoint rule (6.27).
Exercise 6.2 Consider the linearly implicit Euler (LIE) discretization for
the ODE system y = f (y), which reads (for k = 0, 1, . . .)
Exercises
313
yk+1 = yk + (I − τ A)−1 f (yk ) ,
where A = fy (yk ). This scheme is usually monitored to run in some ‘neighborhood’ of the implicit Euler (IE) discretization
F (y) = y − yk − τ f (y) = 0 .
For this purpose the LIE is interpreted as the first iterate of IE and local
contraction within that IE scheme is required. Most LIE codes realize this
requirement via an error oriented criterion introduced in [66]. Here we want
to look at a residual based variant due to [120].
a) On the basis of the residual based Newton-Mysovskikh theorem derive a
computational monitor that is cheap to evaluate.
b) Of which order O(τ s ) is this contraction factor? Derive an adaptive stepsize procedure on that basis.
Hint: Interpret the method as a continuation method with embedding
parameter τ .
c) Compare the error oriented and the residual based variant in terms of
computational amount per discretization step.
d) Optional: Implement the two variants within an adaptive integrator (like
LIMEX) and compare them at several ODE examples.
Exercise 6.3 Consider the system of n nonlinear differential equations
(with time variable t)
ẋ = F (x) , x(0) = x0
modeling some process x(t). Assume that there exists a dynamical invariant
(such as mass conservation) of the form
eT x(t) = eT x0 , eT = (1, . . . , 1) ∈ Rn .
In many cases, one is only interested in a steady state solution x∗ = x(∞)
defined by
F (x∗ ) = 0 .
Since, in general, x∗ will depend on the initial value x0 , uniqueness of the
solution is not guaranteed.
a) Show that the Jacobian F (x∗ ) is singular, which makes a naive application of Newton methods impossible.
b) As a remedy, consider the iterative method
. k /
.
/
F (x )
F (xk )
k
Δx
=
−
,
xk+1 := xk + Δxk
eT
0
314
6 Stiff ODE Initial Value Problems
started by some initial guess x0 . Show that the thus produced iterates
satisfy
e T xk = e T x0 .
What kind of restriction is necessary for the choice of x0 ?
c) Develop a program to treat the described problem type—test examples
may come from chemical kinetics, where mass conservation is often tacitly
assumed without explicitly stating it.
Exercise 6.4 Consider the pseudo-transient continuation method with approximate Jacobian A ≈ F (x0 ). Upon using the notation of Section 6.4.1
and, in addition,
(A − F (x0 ))u ≤ δ|ν|u ,
δ < 1,
prove a variant of Theorem 6.6, containing results on the residual descent
and the optimal pseudo-timestep.
Check: For ν < 0, the optimal timestep τopt comes out to be
τopt =
(1 − δ)|ν|
L0 L2 − (1 − δ)ν 2
in the terms defined—assuming, of course, that the denominator is positive.
Exercise 6.5 How can the iterative linear solver GMRES be optimally
adapted to pseudo-transient continuation? Design a special version, which
saves computing time and storage.
7 ODE Boundary Value Problems
In this chapter, we consider two-point boundary value problems (BVPs) for
ordinary differential equations (ODEs)
y = f (y) , f ∈ C 2 ,
r(y(a), y(b)) = 0 , r ∈ C 2 ,
wherein both the right side f (autonomous for ease of writing) and the boundary conditions r are of dimension n and may be nonlinear. Algorithms for
the solution of such problems can be grouped in two classes: initial value
methods and global discretization methods. The presentation and notation
here closely relates to Chapter 8 in the textbook [71].
Initial value methods. This kind of methods transforms the BVP into
a sequence of initial value problems (IVPs), which are solved by means of
numerical integrators. The most prominent method of this type is the multiple
shooting method, which is a good choice only for problems, wherein a wellconditioned IVP direction exists, i.e. for so-called timelike BVPs. The name
comes from the fact that in this problem class the independent variable t
typically represents a time (or time related) variable. As a rule, there exists
no generalization to boundary value problems for partial differential equations
(PDEs).
Global discretization methods. Conceptually, this kind of BVP methods
does not depend on any preferable direction and is therefore also applicable to
cases, where a well-conditioned IVP direction does not exist, i.e. to so-called
spacelike boundary value problems. In this type of BVP the independent variable t typically represents a space (or space related) variable, which implies
that a generalization to BVPs for PDEs is possible. Such methods include,
e.g., finite difference and collocation methods.
In Section 7.1, the realization of Newton and discrete continuation methods
within the standard multiple shooting approach is elaborated. Gauss-Newton
methods for parameter identification in ODEs are discussed in Section 7.2.
For periodic orbit computation, Section 7.3 presents Gauss-Newton methods,
both in the shooting approach (Sections 7.3.1 and 7.3.2) and in a collocation
approach based on Fourier series (Galerkin-Urabe method in Section 7.3.3).
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance
and Adaptive Algorithms, Springer Series in Computational Mathematics 35,
DOI 10.1007/978-3-642-23899-4_7, © Springer-Verlag Berlin Heidelberg 2011
315
316
7 ODE Boundary Value Problems
In Section 7.4 we concentrate on polynomial collocation methods, which have
reached a rather mature status including affine covariant Newton methods. In
Section 7.4.1, the possible discrepancy between discrete and continuous solutions is studied including the possible occurrence of so-called ‘ghost solutions’
in the nonlinear case. On this basis, the realization of quasilinearization seems
to be preferable in combination with collocation. The following Section 7.4.2
is then devoted to the key issue that quasilinearization can be interpreted
as an inexact Newton method in function space: the approximation errors in
the infinite dimensional setting just replace the inner iteration errors arising
in the finite dimensional setting. With this insight, an adaptive multilevel
control of the collocation errors can be realized to yield an adaptive inexact
Newton method in function space—which is the bridge to adaptive Newton
multilevel methods for PDEs (compare Section 8.3).
Bibliographical Note. Affine invariant global Newton methods—now
called affine covariant Newton methods—have first been developed in the
frame of multiple shooting techniques by P. Deuflhard [60, 62, 61]. Therein
they have turned out to be of crucial importance for the overall performance,
especially in challenging real life optimal control problems. These Newton
techniques have then quickly been adopted by U.M. Ascher and R.D. Russell
within their adaptive collocation methods [8]—with comparable success, see
also their textbook [9]. They have also played an important role within parameter identification algorithms and their convergence analysis as worked
out by H.G. Bock [29, 31, 32] since 1981.
7.1 Multiple Shooting for Timelike BVPs
In this approach the interval [a, b] is subdivided into a partition
Δ = {a = t1 < t2 < · · · < tm = b},
m > 2.
Let xj ∈ Rn , j = 1, . . . , m denote estimates of the unknown values at the
nodes tj . Then, in terms of the flow Φ, we may define those m − 1 subtrajectories
yj (t) = Φt,tj xj ,
t ∈ [tj , tj+1 ] ,
j = 1, . . . , m − 1
that solve (m−1) independent IVPs. The situation is illustrated in Figure 7.1.
For the solution of the problem the sub-trajectories have to be joined continuously and hence at the intermediate nodes the n continuity conditions
Fj (xj , xj+1 ) = Φtj+1 ,tj xj − xj+1 = 0,
have to hold.
j = 1, . . . , m − 1
7.1 Multiple Shooting for Timelike BVPs
317
In addition, we have to satisfy the n boundary conditions
Fm (x1 , xm ) = r(x1 , xm ) = 0.
ξ3
y(t)
Φt2 t1 ξ1
Φt4 t3 ξ3
ξ2
ξ1
Φt3 t2 ξ2
t1 = a
t2
t
b = t4
t3
Fig. 7.1. Multiple shooting (m = 4).
The overall full nm-dimensional system is written in the form
⎛
⎞
⎛
⎞
x1
F1 (x1 , x2 )
⎜
⎟
⎜
⎟
..
x = ⎝ ... ⎠ ∈ Rn·m , F (x) = ⎝
⎠ = 0.
.
xm
(7.1)
Fm (x1 , xm )
This nonlinear system has a cyclic block structure as indicated in Figure 7.2.
For the solution of the above cyclic nonlinear system (7.1) we compute the
ordinary Newton correction as usual by solving the linear system
F (xk )Δxk = −F (xk ) ,
xk+1 = xk + Δxk ,
k = 0, 1, . . . .
The corresponding Jacobian matrix has the cyclic block structure
⎡
⎤
G1 −I
⎢
⎥
..
..
⎢
⎥
.
.
J = F (x) = ⎢
⎥.
⎣
Gm−1 −I ⎦
A
B
Herein the matrices A, B are the derivatives of the boundary conditions r
with respect to the boundary values (x(a), x(b)) = (x1 , xm ). The propagation
matrices Gj on each of the sub-intervals, also called Wronskian matrices, read
Gj =
∂Φtj+1 ,tj xj
,
∂xj
j = 1, . . . , m − 1 .
318
7 ODE Boundary Value Problems
F1
ξ1
ξ2
F2
Fm = r
ξm
ξ3
Fm−1
ξm−1
Fig. 7.2. Cyclic system of nonlinear equations
7.1.1 Cyclic linear systems
The block structure of the Jacobian matrix gives rise to a block cyclic linear
system of the following kind:
G1 Δx1
−Δx2
..
= −F1
..
.
.
Gm−1 Δxm−1
AΔx1
−Δxm
= −Fm−1
+BΔxm
= −Fm = −r .
If this linear system were just solved by some (sparse) direct elimination
method, then global Newton methods as described in the preceding sections
could be directly taken off the shelf.
For timelike BVPs, however, there exists an efficient alternative option, which
opens the door to the construction of interesting specific Gauss-Newton methods. This option dates back to a suggestion of J. Stoer and R. Bulirsch [187].
It is often called condensing algorithm, since it requires only the decomposition of a ‘condensed’ (n, n)-matrix E instead of that of the total Jacobian
(nm, nm)-matrix J. In order to convey the idea, we present the idea first for
the case m = 3:
(1) G1 Δx1
−Δx2
(2)
G2 Δx2
(3) AΔx1
= −F1
−Δx3
= −F2
+BΔx3
= −r .
First we multiply (1) by G2 from the left and add the result
G2 G1 Δx1 − G2 Δx2 = −G2 F1
7.1 Multiple Shooting for Timelike BVPs
319
to equation (2). This gives
G2 G1 Δx1 − Δx3 = −(F2 − G2 F1 ) ,
which after multiplication by B yields
BG2 G1 Δx1 − BΔx3 = −B(f2 − G2 F1 ).
Finally, by addition of equation (3) it follows that
(A + BG2 G1 ) Δx1 = −r − B(F2 − G2 F1 ) .
<
=>
?
<
=>
?
=−u
=E
Hence, in the general case m ≥ 2 we obtain the following algorithm:
a)
Evaluate by recursion over j = 1, . . . , m − 1
E := A + BGm−1 · · · G1 ,
u := r + B [Fm−1 + Gm−1 Fm−2 + · · · + Gm−1 . . . G2 F1 ] .
b)
Solve the linear (n, n) − system
(7.2)
EΔx1 = −u.
c)
Execute the explicit recursion
Δxj+1 := Gj Δxj + Fj ,
j = 1, . . . , m − 1.
The memory required by this algorithm is essentially m · n2 . The computational cost is dominated by the accumulation of the matrix E as an (m − 1)fold product of (n, n)-matrices. Together with the decomposition of E this
results in a cost of O(m · n3 ) operations, where terms of order O(n2 ) have
been neglected as usual.
The large sparse Jacobian matrix J and the small matrix E are closely connected as can be seen by the following lemma.
Lemma 7.1 Notation as just introduced. Define Wj = Gm−1 · · · Gj and
E := A + BW1 . Then
det(J) = det(E) .
(7.3)
Moreover, if E is nonsingular, one has the decomposition
LJR = S,
in terms of the block matrices
J −1 = RS −1 L
(7.4)
320
7 ODE Boundary Value Problems
⎡
BW2 . . .
⎤
B, I
⎢
⎢ −I
⎢
L := ⎢
⎢
..
⎢
.
⎣
−I,
⎥
⎥
⎥
⎥,
⎥
⎥
⎦
⎡
R−1
⎢
⎢ −G1 , I
⎢
:= ⎢
⎢
⎢
⎣
.. ..
. .
⎥
⎥
⎥
⎥,
⎥
⎥
⎦
−Gm−1 , I
0
S := diag(E, I, . . . , I),
⎤
I
S −1 = diag(E −1 , I, . . . , I) .
Proof. The decomposition (7.4) is a direct formalization of the above block
Gaussian elimination. The determinant relation (7.3) follows from
det(L) = det(R) = det(R−1 ) = 1 .
With
det(J) = det(S)
the proof is completed.
Interpretation. The matrix E is an approximation of the special sensitivity
matrix
∂r
E(a) =
= A + BW (b, a)
∂ya
corresponding to the BVP as a whole. Herein W (·, ·) denotes the propagation
matrix of the variational equation. Generically this means that, whenever the
underlying BVP has a locally uniqueness solution, a locally unique solution
x∗ = (x∗1 , . . . , x∗m ) is guaranteed—independent of the partitioning Δ.
Separable linear boundary conditions. This case arises when part of the
boundary conditions fix part of the components of x1 at t = a and part of
the components of xm at t = b. The situation can be conveniently described
in terms of certain projection matrices Pa , P̄a , Pb , P̄b such that
P̄a A = Pa ,
P̄a B = 0 ,
rank(P̄a ) = rank(Pa ) = na < n ,
P̄b B = Pb ,
P̄b A = 0 ,
rank(P̄b ) = rank(Pb ) = nb < n
with na + nb ≤ n. Of course, we will choose initial guesses x01 , x0m for the
Newton iteration so that the separable boundary conditions
P̄a r = 0 , P̄b r = 0
7.1 Multiple Shooting for Timelike BVPs
321
automatically hold. Then the linearization of these conditions
AΔx1 + BΔxm = −r
directly implies
Pa Δx1 = 0 , Pb Δxm = 0 .
Consequently, the variables Pa x1 and Pb xm can be seen to satisfy
Pa x1 = Pa x01 , Pb xm = Pb x0m
throughout the iteration. This part can be realized independent of any elimination method by carefully analyzing the sparsity pattern of the matrices A
and B within the algorithm. As a consequence, the sensitivity matrix E also
has the projection properties
P̄a E = 0,
P̄b E = 0,
EPa = 0,
EPb = 0 .
Iterative refinement sweeps. The block Gaussian elimination technique
(7.2) seems to be highly efficient in terms of memory and computational cost.
A closer look on its numerical stability, however, shows that the method
becomes sufficiently robust only with the addition of some special iterative refinement called iterative refinement sweeps. We will briefly sketch
this technique and work out its consequences for the construction of Newton and Gauss-Newton methods—for details see the original paper [70] by
P. Deuflhard and G. Bader.
Let ν = 0, 1, . . . be the indices of the iterative refinement steps. In lieu
of the exact Newton corrections Δxj the block Gaussian elimination will
supply certain error carrying corrections Δx̃νj so that iterative refinement
will produce nonvanishing differences
dxνj ≈ Δx̃ν+1
− Δx̃νj .
j
In the present framework we might first consider the following algorithm:
ν
ν
(a) duν = drν + B dFm−1
+ Gm−1 dFm−2
+ · · · + Gm−1 · · · G2 dF1ν ,
(b) Edxν1 = −duν ,
(c)
dxνj+1 = Gj dxνj + dFjν j = 1, . . . , m.
However, as shown by the detailed componentwise round-off error analysis in
[70], this type of iterative refinement is only guaranteed to converge under
the sufficient condition
ε(m − 1)(2n + m − 1)κ[a, b] 1 ,
wherein ε denotes the relative machine precision and κ[a, b] the IVP condition
number over the whole interval [a, b]—to be associated with single instead
322
7 ODE Boundary Value Problems
of multiple shooting. This too restrictive error growth can be avoided by a
modification called iterative refinement sweeps. As before, this modification
also begins with an implementation of iterative refinement for the ‘condensed’
linear system
EΔx̃1 + u ≈ 0 .
Suppose, for the time being, that
dx̃1 ≤ eps,
where eps is the relative tolerance prescribed for the Newton iteration. Then
some sweep-index jν ≥ 1 can be defined such that
dx̃νj ≤ eps,
j = 1, . . . , jν .
If we now set part of the residuals deliberately to machine-zero, say,
dFjν = 0,
j = 1, . . . , jν − 1 ,
then this modified iterative refinements process can be shown to converge
under the less restrictive sufficient condition
ε(m − 1)(2n + m − 1)κΔ [a, b] < 1 ,
wherein now the quantity κΔ [a, b] enters, which denotes the maximum of the
IVP condition numbers on each of the subintervals of the partitioning Δ.
Obviously, this quantity reflects the IVP condition number to be naturally
associated with multiple shooting. Under this condition it can be shown that
jν+1 ≥ jν + 1 ,
hence the process terminates, at the latest, after m − 1 refinement sweeps.
Whenever the above excluded case j0 = 0 occurs, the iterative refinement
cannot even start. This occurrence does not necessarily imply that the BVP
as such is ill-conditioned—for a detailed discussion of this aspect see again
the textbook [71].
Rank reduction. The iterative refinement sweeps cheaply supply a condition number estimate for the sensitivity matrix E via
cd(E) =
dx̃01 ≤ cond(E) .
Δx̃01 ε
Even without iterative refinement sweeps a cheap condition number estimate
may be available: Assume that separable boundary conditions have been
split off via the above described projection. Let E denote the remaining part
of the sensitivity matrix which is then treated by QR-decomposition with
7.1 Multiple Shooting for Timelike BVPs
323
column pivoting—for details see, e.g., [77, Section 3.2.2]. In this setting the
subcondition number
sc(E) ≤ cond(E)
is easily computable.
If either
ε cd(E) ≥
⇐⇒
1
2
dx̃01 ≥ 12 Δx̃01 ,
which is equivalent to j0 = 0 or
ε sc(E) ≥
1
2
,
ε cond(E) ≥
1
2
then
.
In other words: in either case E is rank-deficient and the condensed system
is ill-conditioned. In this situation we may replace the condensed equation
EΔx1 = −u
by the underdetermined linear least squares problem
EΔx1 + u2 = min .
This linear system may be ‘solved’ by means of the Moore-Penrose pseudoinverse as
Δx1 = −E + u .
Upon leaving the remaining part of the condensing algorithm unaltered, the
thus modified elimination process can be formally described by some generalized inverse
J − = RS + L
(7.5)
with R,S,L as defined in Lemma 7.1 and
⎡ +
E
⎢
I
⎢
S+ = ⎢
..
⎣
.
⎤
⎥
⎥
⎥.
⎦
I
As can be easily verified (see Exercise 4.7), this generalized inverse is an outer
inverse and can be uniquely defined by the set of four axioms
(J − J)T
= (RRT )−1 J − J(RRT ) ,
(JJ − )T
= (LT L)JJ − (LT L)−1 ,
J − JJ −
= J− ,
JJ − J
= J.
(7.6)
This type of generalized inverse plays a role in a variety of more general
BVPs, some of which are given in the subsequent Sections 7.2 and 7.3.
324
7 ODE Boundary Value Problems
7.1.2 Realization of Newton methods
On the basis of the preceding sections we are now ready to discuss the actual
realization of global Newton methods within multiple shooting techniques for
timelike BVPs.
Jacobian matrix approximations. In order to establish the total Jacobian
J, we must approximate the boundary derivative matrices A, B and the
propagation matrices G1 , . . . , Gm−1 .
Boundary derivatives. Either an analytic derivation of r (not too rare case)
or a finite difference approximation
. δr
. δr
A=
, B=
δx1
δxm
will be realized.
Propagation matrices. The propagation matrices Gj are also called Wronskian
matrices. Whenever the derivative matrix fy (y) of the right side is analytically
available, then numerical integration of the n variational equations
Gj = fy (y(t))Gj , Gj (tj ) = In
(7.7)
might be the method of choice to compute them. If fy is not available analytically, then some internal differentiation as suggested by H.G. Bock [31, 32]
should be applied—see also [71, Section 8.2.1]. Its essence is a numerical
differencing of the form
. δf (y)
fy (y) =
,
δy
which then enters into the numerical solution of discrete variational equations
instead of (7.7). The actual realization of this idea requires special variants
of standard integration software [112, 31, 32]. Note that any such approach
involves, of course, the simultaneous numerical integration of y = f (y(t)) to
obtain the argument y(t) in fy .
Scaling. Formally speaking, each variable xj will be transformed as
xj −→ Dj−1 xj ,
wherein the diagonal matrices Dj > 0 represent some carefully chosen scaling.
Formal consequences are then
Fj
−1
−→ Dj+1
Fj ,
Gj
−1
−→ Dj+1
Gj Dj =: Ĝj .
In actual computation, this means replacing
Fj −1
−→ Dj+1
Fj ,
Δxj −→
Dj−1 Δxj .
(7.8)
7.1 Multiple Shooting for Timelike BVPs
325
Inner products and norms. In view of the underlying BVP we may
want to modify the Euclidean inner product and norm by including information about the mesh Δ = {t1 , . . . , tm }. For example, let (·, ·) denote some
(possibly scaled) Euclidean inner product for the vectors u = (u1 , . . . , um ),
v = (u1 , . . . , vm ), uj , vj ∈ Rn . Then we may define some discrete L2 -product
by virtue of
|b − a|(u, v)Δ
=
(u1 , v1 )|t2 − t1 | + (um , vm )|tm − tm−1 |
m−1
+ j=2 (uj , vj )|tj+1 − tj−1 |
(7.9)
and its induced discrete L2 -norm as
(u, u)Δ ≡ u2Δ .
Quasi-Newton updates. Any approximation of the Wronskian matrices
Gj requires a computational cost of ∼ n trajectory evaluations. In order to
save computing time per Wronskian evaluation, we may apply rank-1 updates
as long as the iterates remain within the Kantorovich domain around the
solution point—i.e., when the damping strategies in Section 2.1.4 supply
λk = λk−1 = 1 .
Of course, the sparse structure of the total Jacobian J must be taken into
account. We assume the boundary derivative approximations A and B as
fixed. Then the secant condition (1.17) for the total Jacobian
(Jk+1 − Jk )Δxk = F (xk+1 )
splits into the separate block secant conditions
(Gk+1
− Gkj )Δxkj = Fj (xk+1
, xk+1
j
j
j+1 ) ,
j = 1, . . . , m − 1 .
Upon applying the ideas of Section 2.1.4, we arrive at the following rank-1
update formula:
Gk+1
= Gkj + Fj (xk+1
, xk+1
j
j
j+1 )
(Δxkj )T
Δxkj 22
,
j = 1, . . . , m − 1 .
(7.10)
In a scaled version of the update formula (7.10), we will either update the Ĝj
from (7.8) directly or, equivalently, update Gj replacing
(Dj−2 Δxj )T
ΔxTj
−→
Δxj 22
Dj−1 Δxj 22
in the representation (7.10). As worked out in detail in Section 2.1.4 above,
scaling definitely influences the convergence of the corresponding quasiNewton iteration (compare also [59, Section 4.2]).
326
7 ODE Boundary Value Problems
Adaptive rank strategy. Assume that the condensed matrix E has been
indicated as being ‘rank-deficient’. In this case we need not terminate the
Newton iteration, but may continue by an intermediate Gauss-Newton step
with a correction of the form
Δxk = −F (xk )− F (xk ) .
Upon recalling Section 4.1.1, the generalized inverse F (x)− can be seen to
be an outer inverse. Therefore Theorem 4.7 guarantees that the thus defined
ordinary Gauss-Newton iteration converges locally to a solution of the system
Fj (xj , xj+1 ) = 0 , j = 1, . . . , m − 1 ,
(7.11)
r(x1 , xm )2 = min .
The modified trust region strategies of Section 4.3.5 can be adapted for the
special projector
P := J − J .
Note, however, that intermediate rank reductions in this context will not
guarantee an increase of the feasible damping factors (compare Lemma 4.17 or
Lemma 4.18), since P is generically not orthogonal (for m > 2). Nevertheless
a significant increase of the damping factors has been observed in numerical
experiments.
Obviously, the thus constructed Gauss-Newton method is associated with an
underdetermined least squares BVP of the kind
y = f (y) ,
r(y(a), y(b))2 = min .
Level functions. In the rank-deficient case, the residual level function
m
T (x|I) = F (x)2 =
Fj (x)2
j=1
no longer has the Gauss-Newton correction Δx = −F (x)− F (x) as a descent
direction. Among the practically interesting level functions, this property still
holds for the above used natural level function T (x|J − ) or for the hybrid level
function
T (x|R−1 J − ) = R−1 J − F (x)2 = Δx1 2 +
m−1
Fj (x)2 .
j=1
The proof of these statements is left as Exercise 7.4.
7.1 Multiple Shooting for Timelike BVPs
327
Bibliographical Note. The affine covariant Newton method as described
here is realized, e.g., in the multiple shooting code BVPSOL due to P. Deuflhard
and G. Bader [70] and the optimal control code BOUNDSCO due to J. Oberle
[162]. Among these only BVPSOL realizes the Gaussian block elimination (Section 7.1.1) including the rank-deficient option with possible intermediate
Gauss-Newton steps. Global sparse solution of the cyclic linear Newton systems is implemented in the code BOUNDSCO and as one of two options in
BVPSOL; a rank-strategy is not incorporated within global elimination.
7.1.3 Realization of continuation methods
Throughout this section we consider parameter dependent two-point boundary value problems of the kind
y = f (y, τ ) ,
r(y(a), y(b), τ ) = 0 ,
which give rise to some parameter dependent cyclic system of nonlinear equations
F (x, τ ) = 0 .
Typical situations are that either the τ -family of BVP solutions needs to
be studied or a continuation method is applied to globalize a local Newton method. Generally speaking, the parameter dependent mapping F is
exactly the case treated in Section 5. Hence, any of the continuation methods described there can be transferred—including the automatic control of
the parameter stepsizes Δτ .
Newton continuation methods. Assume the BVP under consideration
has no turning or bifurcation points—known either from external insight
into the given scientific problem or from an a-priori analysis. Then Newton
continuation methods as presented in Section 5.1 are applicable.
Classical continuation method. This algorithm (of order p = 1) deserves no
further explanation. All the details of Section 5.1 carry over immediately.
Tangent continuation method. For this algorithm (of order p = 2) we need to
solve the linear system
˙ ) = Fτ (x, τ ) ,
Fx (x, τ )x̄(τ
which is the same type of block cyclic linear system as for the Newton corrections. The above right hand term Fτ (x, τ) can be computed by numerical
integration of the associated variational equations or by internal numerical
differentiation (cf. [31, 32] or [71, Section 8.2.1]).
328
7 ODE Boundary Value Problems
Continuation via trivial BVP extension. A rather popular trick is to
just extend the standard BVP such that
y = f (y, τ ),
τ = 0 ,
r(y(0), y(T ), τ ) = 0,
h(τ ) = 0
(7.12)
with h (τ ) = 0. Any BVP solver applied to this extended BVP then defines
some extended mapping
F (x, τ) = 0,
h(τ ) = 0 .
(7.13)
Let Δxk denote the Newton correction for the equations with fixed τ . Then
the Newton correction (Δz k , Δτ ) for (7.13) turns out to be
˙ k ),
Δz k = Δxk + Δτ k x̄(τ
Δτ k = −
h(τ k )
.
h (τ k )
(7.14)
The proof of this connection is left as Exercise 7.1. If one selects τ 0 such
that h(τ 0 ) = 0, then the extension (7.12) realizes a mixture of continuation
methods of order p = 1 and p = 2. An adaptive control of the stepsizes Δτ
here arises indirectly via the damping strategy of Newton’s method.
Example 7.1 Space shuttle problem. This optimal control problem stands
for a class of highly sensitive BVPs from space flight engineering. The underlying physical model (very close to realistic) is due to E.D. Dickmanns
[87]. The full mathematical model has been documented in [81]. The stated
mathematical problem is to find an optimal trajectory of the second stage of
a Space Shuttle such that a prescribed maximum permitted skin temperature
of the front shield is not exceeded. The real problem of interest is a study with
respect to the temperature parameter, say τ . For technological reasons, the
aim is to drive down the temperature as far as possible. This problem gave
rise to a well-documented success story for error oriented Newton methods
(earlier called affine ‘invariant’ instead of affine covariant).
The unconstrained trajectory goes with a temperature level of 2850◦F (equivalent to τ = 0.072). The original technological objective of NASA had been
optimal flight trajectories at temperature level 2000◦F (equivalent to τ = 0.).
However, the applied continuation methods just failed to continue to temperatures lower than 2850◦ F ! One reason for that failure can already be seen in
the sensitivity matrix: the early optimal control code OPTSOL of R. Bulirsch
[42], improved 1972 by P. Deuflhard [59] (essentially in the direction of error
oriented Newton methods), revealed a subcondition number
sc(E) = 0.2 · 1010
at that temperature. As a consequence, any traditional residual based Newton
methods, which had actually been used at NASA within the frame of classical
7.1 Multiple Shooting for Timelike BVPs
329
continuation, are bound to fail. The reasons for such an expectation have been
discussed in Sections 3.3.1 and 3.3.2.
In 1973, H.-J. Pesch attacked this problem by means of OPTSOL, which in
those days still contained classical continuation with empirical stepsize control, but already error oriented Newton methods [59] with empirical damping strategy. With these techniques at hand, the first successful continuation
steps to τ < 0.072 were at all possible—nicely illustrating the geometric
insight from Section 3.3.2. However, computing times had been above any
tolerable level, so that H.-J. Pesch eventually terminated the continuation
process at τ = 0.0080 with a final empirical stepsize Δτ = −0.0005. Further
improvements were possible by replacing
• the classical Newton damping strategy by an adaptive one [63], similar to
the adaptive trust region predictor given in Section 3.3.3 and
• the classical continuation method with empirical stepsize selection by their
adaptive counterparts as presented in Section 5.1.
For the last continuation step performed by H.-J. Pesch, Table 7.1 shows
the comparative computational amount for different continuation methods
(counting full trajectories to be computed within the multiple shooting approach).
Continuation method
classical
Newton method
residual based
work
failure
classical
error oriented,
empirical damping
∼ 340
classical
error oriented,
adaptive trust region
114
trivial BVP
extension
error oriented,
adaptive trust region
48
tangent
error oriented,
adaptive trust region
18
Table 7.1. Space Shuttle problem: Fixed continuation step from τ = 0.0085 to
τ = 0.0080. Comparative computational amount for different Newton continuation
methods.
In 1975, an adaptive error oriented Newton method [81] in connection with
the trivial BVP extension made it, for the first time, possible to solve the
original NASA problem for temperature level 2000◦F (τ = 0.). Results of
technical interest have been published by E.D. Dickmanns and H.-J. Pesch
[88]. The performance of this computational technique for temperatures even
below the NASA objective value is documented in Table 7.2. As can be seen,
330
7 ODE Boundary Value Problems
this kind of continuation technique, even though it succeeds to solve the
problem, still performs a bit rough.
Continuation
0.0
→
0.0
→
0.0
→
-0.0005 →
-0.0010 →
-0.0020 →
-0.0050 →
-0.0050 →
-0.0050 →
0.0050
→
0.0
→
sequence
-0.0050
-0.0010
-0.0005
-0.0010
-0.0020
-0.0050
-0.0200
-0.0100
-0.0100
-0.0080
-0.0080
work
60 fail
60 fail
80
63
50
65
30 fail
30 fail
67 fail
66
571
Remarks
switching structure totally disturbed
negative argument in log-function
switching structure totally disturbed
as above
Newton method fails to converge
prescribed final parameter
overall amount
Table 7.2. Space Shuttle problem: Adaptive continuation method [81] via trivial BVP extension (7.12).
A much smoother and faster behavior occurs when adaptive tangent continuation as worked out in Section 5.1 is applied—just see Table 7.3. With this
method the temperature could be lowered even down to 1700◦ F . Starting
from these data, H.G. Bock [30] computed an achievable temperature of only
890◦ C from the multiple shooting solution of a Chebyshev problem assuming
that all state constraints of the problem are to be observed.
Continuation sequence
work Remarks
0.0
→ -0.0035
47
ordinary Newton method
-0.0035 → -0.0057
32
throughout the computation;
-0.0057 → -0.0080a 31
switching structure never disturbed
0.0
→ -0.0080
110
overall amount
a
stepsize cut off to prescribed final value τ = −0.0080.
Table 7.3. Space Shuttle problem: Adaptive tangent continuation [61]. See also
Section 5.1 here.
Remark 7.1 It may be interesting to hear that none of these ‘cooler’ space
shuttle trajectories has been realized up to now. In fact, the author of this
book has presented the optimal 2000◦ F trajectories in 1977 within a seminar
at NASA, Johnson Space Flight Center, Houston; the response there had
been that the countdown for the launching of the first space shuttle (several
7.1 Multiple Shooting for Timelike BVPs
331
years ahead) had already gone too far to make any substantial changes. The
second chance came when Europe thought about launching its own space
shuttle HERMES; in fact, the maximum skin temperature assumed therein
turned out to be the same as for the present NASA flights! Sooner or later, a
newcomer in the space flight business (and this is big business!) will exploit
this kind of knowledge which permits him (or her) to build a space shuttle
in a much cheaper technology.
Gauss-Newton continuation method. As soon as turning or bifurcation
points might arise, any Newton continuation method is known to be inefficient and Gauss-Newton techniques come into play—compare Section 5.2.
The basic idea behind these techniques is to treat the parameter dependent
nonlinear equations as an underdetermined system in terms of the extended
variable z = (x, τ ) = (x1 , . . . , xm , τ ). In addition to the Wronskian approximations Gj we therefore need the derivatives
gj :=
∂Φtj +1,tj (τ )xj
,
∂τ
j = 1, . . . , m .
With these definitions the Jacobian (nm, nm + 1)-matrix now has the block
structure
⎡
⎤
G1 −I
g1
⎢
.. ⎥
..
..
⎢
.
.
. ⎥
J =⎢
⎥.
⎣
Gm−1 −I gm−1 ⎦
A
B
gm
Based on this structure, Gaussian block elimination offers a convenient way
to compute a Gauss-Newton correction
0 = −J − F
Δz
in terms of the outer inverse J − already introduced in (7.5). The computation
0 can be conveniently based on a QR-decomposition of the (n, n + 1)of Δz
matrix [E, g], where
E
g
:=
:=
A + BGm−1 · · · · · G1 ,
gm + B(gm−1 + · · · + Gm−1 · · · · · G2 g1 ) .
With this we obtain a variant of the condensing algorithm
@1
Δx
= −[E, g]+u ,
0
Δτ
0 j+1
0 j + gj Δτ
0 + Fj , j = 1, . . . , m − 1 .
Δx
= Gj Δx
(7.15)
Of course, iterative refinement sweeps must be properly added, see Section
7.1.1. This kind of Gauss-Newton continuation would need to be coupled by
332
7 ODE Boundary Value Problems
some extra step-size control in the continuation parameter τ —which is not
worked out here.
Instead we advocate a realization of the standard Gauss-Newton continuation
method as developed in Section 5.2. In the spirit of (5.28) and (5.29), a GaussNewton correction in terms of the Moore-Penrose pseudo-inverse can be easily
computed via
0
0 − (t, Δz) t ,
Δz = −J + F = Δz
(7.16)
(t, t)
wherein t now denotes any kernel vector satisfying Jt = 0. Let t =
(t1 , . . . , tm , tτ ) denote the partitioning of a kernel vector. Then components
(t1 , tτ ) can be computed from the (n, n + 1)-system
Et1 + gtτ = 0
again via the QR-decomposition. The remaining components are once more
obtained via explicit recursion as
tj+1 = Gj tj + gj tτ , j = 1, . . . , m − 1 .
0 and the kernel vector t finally yields
Insertion of the particular correction Δz
the Gauss-Newton correction Δz. The local convergence analysis as well as
the corresponding step-size control in τ can then be copied from Section 5.2
without further modification.
As for the inner products arising in the above formula, the discrete L2 -product
(·, ·)Δ defined in (7.9) looks most promising, since it implicitly reflects the
structure of the BVP.
Detection of critical points. Just as in Section 5.2.3, certain determinant
pairs need to be computed. This is especially simple in the context of the
QR-decomposition of the matrix [E, g] in (7.15).
Bibliographical Note. More details are given in the original paper [73]
by P. Deuflhard, B. Fiedler, and P. Kunkel. There also a performance comparison of MULCON, a multiple shooting code with Gauss-Newton continuation
as presented here, and AUTO, a collocation code with pseudo-arclength continuation due to E. Doedel [89] is presented: the given numerical example
nicely shows that the empirical pseudo-arclength continuation is robust and
reliable, but too cautious and therefore slower, whereas the adaptive GaussNewton continuation is also robust and reliable, but much faster. Moreover,
continuous analogs of the augmented system of G. Moore for the characterization of bifurcations are worked out therein.
7.2 Parameter Identification in ODEs
333
7.2 Parameter Identification in ODEs
This section deals with the inverse problem in ordinary differential equations
(ODEs): given a system of n nonlinear ODEs
y = f (y, p) , y(0) given , p ∈ Rq ,
determine the unknown parameter vector p such that the solution y(t, p) ‘fits’
given experimental data
zj ∈ Rn , j = 1, . . . , M .
(τj , zj )
Let
δy(τj , p) := y(τj , p) − zj , j = 1, . . . , M .
denote the pointwise deviations between model and data with prescribed
statistical tolerances δzj , j = 1, . . . , M . If some of the components of zj are
not available, this formally means that the corresponding components of δzj
are infinite. In nonlinear least squares, the deviations are measured via a
discrete (weighted) l2 -product (·, ·), which leads to the problem
(δy, δy) =
1
M
M
Dj−1 δy(τj , p)22 = min
j=1
with diagonal weighting matrices
Dj := diag(δzj1 , . . . , δzjn ) , j = 1, . . . , M .
If we define some nonlinear mapping F by
⎡
⎤
D1−1 δy(τ1 , p)
⎢
⎥
..
F (p) := ⎣
⎦,
.
−1
DM δy(τM , p)
then our least squares problem reads
F (p)22 ≡ (δy, δy) = min .
If all components at every data point τj have been measured (rare case), then
F : Rq −→ RL with L = nM . Otherwise some L < nM occurs.
We are thus guided to some constrained nonlinear least squares problem
y = f (y, p),
(δy, δy) = min ,
where the ODEs represent the equality constraints. This problem type leads
to a modification of the standard multiple shooting method.
334
7 ODE Boundary Value Problems
The associated Jacobian (L, q)-matrix F (p) must also be computed exploiting its structure numerically
⎡ −1
⎤
D1 yp (τ1 )
⎢
⎥
..
F (p) = ⎣
(7.17)
⎦.
.
Dl−1 yp (τl )
Herein the sensitivity matrices yp each satisfy the variational equation
yp = fy (y, p)yp + fp (y, p)
with initial values yp (tj ) = 0, tj ∈ Δ. Of course, we will naturally pick m
multiple shooting nodes out of the set of M measurement nodes, which means
that
Δ := {t1 , . . . , tm } ⊆ {τ1 , . . . , τM }
with, in general, m M . As before, sub-trajectories Φt,tj (p)xj are defined
per each subinterval t ∈ [tj , tj+1 ] via the initial value problem
y = f (y, p) , y(tj ) = xj .
Figure 7.3 gives a graphical illustration of the situation for the special case
M = 13, m = 4.
y(t)
ξ3
Φt2 t1 ξ1
Φt4 t3 ξ3
ξ2
ξ1
Φt3 t2 ξ2
τ2 τ3 τ4
t1 = τ 1 = a
τ6 τ7 τ8
t2 = τ 5
τ2 τ3 τ4
t3 = τ 9
t
t4 = τ13 = b
Fig. 7.3. Multiple shooting for parameter identification (M = 13, m = 4).
Unknowns to be determined are (x, p) = (x1 , . . . , xm , p). If we introduce the
convenient notation
⎡
⎤
D1−1 (Φτ1 ,t1 (p)x1 − z1 )
⎢
⎥
..
⎢
⎥
.
r(x1 , . . . , xm , p) := ⎢
⎥,
⎣ D−1 (ΦτM −1 ,tm−1 (p)xm−1 − zM −1 ) ⎦
M −1
−1
DM
(xm − zM )
7.2 Parameter Identification in ODEs
335
we arrive at the parameter identification problem in its multiple shooting
version
Fj (xj , xj+1 , p) := Φtj+1 ,tj (p)xj − xj+1 = 0 , j = 1, . . . , m − 1 ,
r(x1 , . . . , xm , p)22 = min .
Obviously, this is a constrained nonlinear least squares problem with the
continuity equations as nonlinear constraints, an overdetermined extension
of (7.11).
For ease of notation we write the whole mapping as
⎡
⎤
F1 (x1 , x1 , p)
⎢
⎥
..
⎢
⎥
.
F (x, p) = ⎢
⎥
⎣ Fm−1 (xm−1 , xm , p) ⎦
r(x1 , . . . , xm , p)
and its block structured Jacobian matrix (ignoring weighting matrices for
simplicity) as
⎡
⎤
G1 −I
P1
⎢
⎥
..
..
..
⎢
⎥
.
.
.
J=⎢
⎥.
⎣
Gm−1 −I Pm−1 ⎦
B1 . . . Bm−1 Bm
Pm
The above matrices Pj , j = 1, . . . , m represent the parameter derivatives
of the mapping F consisting just as in (7.17) of sensitivity matrices; their
length and initial values depend on the available measurement data and on
the selection of the multiple shooting nodes out of the measurement modes—
details are omitted here, since they require clumsy notation.
Upon recalling (7.11) and (7.5), the corresponding constrained Gauss-Newton
corrections are defined as
(Δxk , Δpk ) = −J(xk , pk )− F (xk , pk )
or, more explicitly, via the block system
Gj Δxj − Δxj+1 + Pj Δp = −Fj , j = 1, . . . , , m − 1 ,
B1 Δx1 + · · · + Bm Δxm + Pm Δp + r22 = min .
Gaussian block elimination. Proceeding as in the simpler BVP case, we
here obtain
Δx1
= −[E, P ]+ u ,
Δp
wherein the quantities E, P, u are computed recursively from
336
7 ODE Boundary Value Problems
P̄m := Pm , B̄m := Bm ,
j = m − 1, . . . , 1 : P̄j := P̄j+1 + B̄j+1 Pj ,
B̄j := Bj + B̄j+1 Gj ,
E := B̄1 , P := P̄1 ,
u := r + Bm [Fm−1 + · · · + Gm−1 · · · · · G2 F1 ] .
The remaining correction components follow from
Δxj+1 = Gj Δxj + Pj Δp + Fj , j = 1, . . . , m − 1 .
In analogy with (7.5) and with the notation for the block matrices L, R
introduced there, the generalized inverse J − can be formally written as
J − = RS − L,
S − = diag ([E, P ]+ , I, · · · , I).
Bibliographical Note. Since 1981, this version of the multiple shooting
method for parameter identification in differential equations has been suggested and driven to impressive perfection by H.G. Bock [29, 31, 32] and his
coworkers. It is implemented in the program PARFIT. A single shooting variant especially designed for parameter identification in large chemical reaction
kinetic networks has been worked out in detail by U. Nowak and the author
[158, 159] in the code PARKIN.
Iterative refinement sweeps. In order to start the iterative refinement
sweeps, we require some iterative correction of the above condensed leastsquares system. A naive iterative correction approach, however, would not
be suitable for large residuals
r̄ = EΔx1 + P Δp + u .
Thus we recommend an algorithm proposed by Å. Bjørck [25] to be adapted
to the present situation. In this approach the above linear least-squares problem is first written in the form of the augmented linear system of equations
−r̄ + EΔx1 + P Δp + u
T
[E, P ] r̄
= 0,
= 0
in the variables Δx1 , Δp, r̄. The iterative correction is then applied to this
system. If it converges—which means that the condensed linear least-squares
problem is regarded as well-posed—then, without any changes, the iterative
refinement sweeps for the explicit recursion can be applied in the same form
as presented in the standard situation treated in Section 7.1.1.
7.3 Periodic Orbit Computation
337
7.3 Periodic Orbit Computation
In this section we are interested in continuous solutions of periodic boundary
value problems:
y = f (y) ,
r(y(0), y(T )) := y(T ) − y(0) = 0
(7.18)
with (hidden) time variable t and (unknown) period T . Here we treat only
the situation that f is autonomous, the nonautonomous case is essentially
standard. In this case f satisfies the variational equation
f = fy · f ,
which can be formally solved as
f (y(t)) = W (t, 0) f (y(0)) ,
(7.19)
wherein W (·, ·) is once more the propagation matrix of the variational equation. Insertion of the periodicity condition y(T ) = y(0) then yields
f (y(T )) = f (y(0)) = W (T, 0)f (y(0)) ,
or, equivalently, with E = E(0) = W (T, 0) − I inserted:
Ef (y(0)) = 0 .
Obviously, the sensitivity matrix E is singular and f (y(0)) is a right eigenvector associated with eigenvalue zero, if only f (y(0)) = 0. The singularity
of E reflects the fact that the phase or time origin is undetermined, causing
a special nonuniqueness of solutions: whenever y(t), t ∈ [0, T ] is a periodic
solution, then y(t + t0 ), t ∈ [0, T ], t0 = 0, is a different periodic solution,
even though it is represented by the same orbit. Obviously, there exists a
continuous solution set generated by S 1 -symmetry. In this situation, we will
naturally aim at computing the orbit directly, which means computing any
trajectory y(t + t0 ), t ∈ [0, T ] without fixing the phase t0 .
In what follows we will assume that the eigenvalue zero of E is simple, which
then implies that
rank[E, f (y(0))] = n
and
ker[E, f ] = (f, 0) .
(7.20)
7.3.1 Single orbit computation
First we treat the case when a single periodic orbit is wanted. In order to
convey the main idea, we start with the derivation of a special Gauss-Newton
method in the framework of single shooting (m = 2).
338
7 ODE Boundary Value Problems
Single Shooting. In this approach one obtains the underdetermined system
of n equations
F (z) = ΦT x − x = 0
in the n + 1 unknowns z = (x, T ) is generated. The corresponding Jacobian
(n, n + 1)-matrix has the form
F (z) = [rx , rT ] = [E, f (x(T ))] ,
or, after substituting a periodic solution,
F (z) = [E, f (x(0))] .
Under the assumption made above the Jacobian matrix has full row rank and
its Moore-Penrose pseudoinverse F (z)+ has full column rank. Hence, instead
of a Newton method we can construct the Gauss-Newton iteration
Δz k = −F (z k )+ F (z),
z k+1 = z k + Δz k ,
k = 0, 1, . . . .
Also under the full rank assumption this iteration will converge locally
quadratically to some solution z ∗ in the ‘neighborhood’ of a starting point
z 0 , i.e., to an arbitrary point on the orbit. This point determines the whole
orbit uniquely.
Multiple shooting. In multiple shooting we need to have fixed nodes. So
we introduce the dimensionless independent variable
s :=
t
∈ [0, 1] .
T
Let
Δ := {0 = s1 < s2 < · · · < sm = 1}
denote the given partitioning with mesh sizes
Δsj := sj+1 − sj , j = 1, . . . , m − 1 .
Then the following conditions must hold
Fj (xj , xj+1 , T ) :=
r(x1 , xm ) :=
ΦT Δsj xj − xj+1 = 0 , j = 1, . . . , m − 1 ,
xm − x1 = 0 .
The subtrajectories can be formally represented by
T Δsj
T Δsj
Φ
xj = xj +
f (y(s))ds .
s=0
With the Wronskian (n, n)-matrices
(7.21)
7.3 Periodic Orbit Computation
Gj
:=
339
∂ΦT Δsj xj
= W (sj+1 , sj ) T (s−s )
j x )
∂xj
Φ
j
and the n-vectors
gj
∂ΦT Δsj xj
= Δsj f (ΦT Δsj xj )
∂T
:=
the underdetermined linear system to be solved in each Gauss-Newton step
has the form
G1 Δx1 −Δx2
..
.
−Δx1
+g1 ΔT =
..
.
..
.
Gm−1 Δxm−1 −Δxm +gm−1 ΔT
+Δxm
=
=
−F1
..
.
−Fm−1
0.
Gaussian block elimination. The ‘condensing’ algorithm as described in Section 7.1.1 will here lead to the small underdetermined linear system
EΔx1 + gΔT + u = 0 ,
where
E = Gm−1 · · · · · G1 − I ,
g := gm−1 + · · · + Gm−1 · · · · · G2 g1 ,
u := Fm−1 + · · · + Gm−1 · · · · · G2 F1 .
At a solution point z ∗ = (x∗1 , . . . , x∗m , T ∗ ) we obtain
E ∗ = W (1, 0) − I ,
g ∗ = f (x∗1 ) ,
just recalling that
f (x∗m ) = W (sm , sj )f (x∗j ) = f (x∗1 ),
Δs1 + · · · + Δsm−1 = 1 .
Hence, the (n, n + 1)-matrix
[E ∗ , g ∗ ] = [E(0), f (y(0))]
has again full row rank n. Consequently, an extension of the single shooting
rank-deficient Gauss-Newton method realizing the Jacobian outer inverse J −
can be envisioned.
Better convergence properties, however, are expected by the standard GaussNewton method which requires the Moore-Penrose pseudoinverse J + of the
total block Jacobian. As in the case of parameter dependent BVPs, we again
compute a kernel vector t = (t1 , . . . , tm , tT ), here according to
340
7 ODE Boundary Value Problems
t1
[E, g]
= 0,
tT
tj+1 = Gj tj + gj tT , j = 1, . . . , m − 1
and combine it with iterative refinement sweeps, of course. The computation
of the actual Gauss-Newton correction then follows from
0
0 − (t, Δz) t ,
Δz = Δz
(t, t)
0 is an arbitrary particular correction vector satisfying
where Δz
0 + F (z k ) = 0 .
F (z k )Δz
Simplified Gauss-Newton method. At the solution point z ∗ , the above
equations lead (up to some normalization factor) to the known solution
(t1 , tT ) = (f (x∗1 ), 0) .
Upon inserting this result into (7.19), we immediately arrive at
tj = f (x∗j ) , j = 2, . . . , m .
This inspires an intriguing modification of the above Gauss-Newton method:
we may insert the ‘iterative’ kernel vector
tkT = 0, tkj = f (xkj ) j = 1, . . . , m
into the expression (7.16). In this way we again obtain some pseudo-inverse
and, in turn, thus define some associated Gauss-Newton method.
Bibliographical Note. The multiple shooting version realizing the Jacobian outer inverse J − has been suggested in 1984 by P. Deuflhard [64] and
implemented in the code PERIOD. The improvement realizing J + has been
proposed in 1994 by C. Wulff, A. Hohmann, and P. Deuflhard[199] and realized in the orbit continuation code PERHOM—for details see the subsequent
Section 7.3.2. The same paper also contains a possible exploitation of symmetry for equivariant orbit problems, following up the work of K. Gatermann
and A. Hohmann [95] for equivariant steady state problems.
7.3.2 Orbit continuation methods
In this section we consider the computation of families of orbits for the parameter dependent periodic boundary value problem
y = f (y, λ) ,
r(y(0), y(T )) := y(T ) − y(0) = 0 ,
7.3 Periodic Orbit Computation
341
where λ is the embedding parameter and T the unknown period. Note that
the S 1 -symmetry now only holds for fixed λ, so that the orbits can be explicitly parametrized with respect to λ. Throughout this section, let {λν } denote
the parameter sequence and
Δλν := λν+1 − λν
the corresponding continuation step sizes to be automatically selected.
Single shooting. In order to convey the main geometrical idea, we again
start with this simpler case. There we must solve the sequence of problems
F (x, T, λ) := ΦT (λ)x − x = 0
for the parameters λ ∈ {λν }.
Classical continuation method. This continuation method, where the previous
orbit just serves as starting guess for the Gauss-Newton iteration to compute
the next orbit, can be implemented without any further discussion, essentially
as described in Section 5.1.
Tangent continuation method. The realization of this method deserves some
special consideration. The Jacobian (n, n + 2)-matrix has the following substructure
[Fx , FT , Fλ ] = [E, f, p]
with E, f as introduced above and p defined as
p :=
∂ΦT (λ)x
.
∂λ
Let t = (tx , tT , tλ ) denote any kernel vector satisfying
Etx + f · tT + p · tλ = 0 .
Under the above assumption that [E, f ] has full row rank n, we here know
that
dim ker[E, f, p] = 2 .
A natural basis for the kernel will be {t1 , t2 } such that
t1 := ker[E, f ] , t2 ⊥ t1 ,
(7.22)
wherein
ti = (tix , tiT , tiλ ) , i = 1, 2 .
From (7.20) and (7.22) we are directly led to the representations (ignoring
any normalization)
t1x = f , t1T = 0 , t1λ = 0
and
342
7 ODE Boundary Value Problems
t2x
t2T
= −[E, f ]+p , t2λ := 1 .
Upon recalling that t1 reflects the S 1 -symmetry of each orbit, tangent continuation will mean to continue along t2 . In the notation of Section 5 this
means that guesses (x̂, T̂ ) can be predicted as
2 x̂(λν+1 ) − x̄(λν )
tx =
2
· Δλν .
t
T̂ (λν+1 ) − t(λν )
T
λν
These guesses are used as starting points for the Gauss-Newton iteration as
described in Section 7.3.1. Since t1T = t1λ = 0, the property t1 ⊥ t2 also
implies t1x ⊥ t2x , which means that
x̂(λν+1 ) − x̄(λν ) ⊥ f (x̄(λν )) .
(7.23)
The geometric situation is represented schematically in Figure 7.4.
f (x̄(τν ))
x̂(τν+1 )
x̄(τν )
H
τν -orbit
Fig. 7.4. Orbit continuation: H Hopf bifurcation point, — stable steady states,
- - - unstable steady states.
Multiple Shooting. As in (7.21) we again deal with fixed nodes
Δ := {0 = s1 < s2 < · · · < sm = 1}
instead of variable nodes tj = sj T, j = 1, . . . , m.
7.3 Periodic Orbit Computation
343
Switching notation, let now t = (t1 , . . . , tm , tT , tλ ) denote a selected kernel
vector satisfying
Gj tj − tj+1 + gj · tT + pj tλ = 0 , j = 1, . . . , m − 1 ,
tm = t1 ,
wherein Gj , gj , and pj are essentially defined as in single shooting (m = 2).
For m > 2, the Jacobian nullspace is still two-dimensional. For continuation
we again choose the tangent vector
t = (f (x1 ), . . . , f (xm ), 0, 0), x = x̄(λν ) .
Gaussian block elimination. In this setting, we will compute
@1
Δx
:= −[E, f ]+ p · Δλν
@
ΔT
with
p := pm−1 + · · · + Gm−1 · · · · · G2 p1
and, recursively, for j = 1, . . . , m − 2:
0 j+1 = Gj Δx
@j + gj ΔT
@ + pj Δλν .
Δx
As a straightforward consequence, we may verify that
@
Δx
m = Δx1 .
The thus defined continuation
@j , j = 1, . . . , m,
x̂j (λν+1 ) − x̄j (λν ) = Δx
@
T̂ (λν+1 ) − t(λν ) = ΔT
clearly satisfies the local orthogonality property
x̂1 (λν+1 ) − x̄1 (λν ) ⊥ f (x̄1 (λν )) ,
which is biased towards the node s1 = 0. Therefore, already from a geometrical point of view, the above continuation method should be modified such
that the global orthogonality
x̂(λν+1 ) − x̄(λν ) ⊥ f (x̄(λν ))
holds as a natural extension of (7.23). This directly leads us to the following
orbit continuation method
0
0 − (fν , Δx) fν
x̂(λν+1 ) − x̄(λν ) = Δx
(fν , fν )
with fν = f (x̄(λν )). Once more, the periodicity condition
x̂m (λν+1 ) = xˆ1 (λν+1 )
can be shown to hold.
344
7 ODE Boundary Value Problems
Hopf bifurcations. The detection of Hopf bifurcation points H is an easy
computational task. First, orbit continuation beyond H is impossible, as long
as we come from the side of the periodic orbits, which we here do—see Figure 7.4. Second, at H, the condition f = 0 leads to rank[E, f ] < n—a behavior that already shows up in a neighborhood of H. Third, the automatic
stepsize control will lead to a significant reduction of the stepsizes Δλν . As
soon as a Hopf bifurcation point seems to close by, its precise computation
can be done switching to the augmented system suggested by A.D. Jepson
[124].
Bibliographical Note. The above orbit continuation has been worked
out in 1994 by C. Wulff, A. Hohmann, and P. Deuflhard [199] and realized in
the code PERHOM. The same paper also covers the detection and computation
of Hopf bifurcations and period doublings. Particular attention is paid to the
computational exploitation of symmetries following up work of M. Dellnitz
and B. Werner [50] and of K. Gatermann and A. Hohmann [95], the latter for
steady state problems only, the former including Hopf bifurcations as well.
7.3.3 Fourier collocation method
In quite a number of application fields the desired periodic solution y to
period T = 2π/ω is just expanded into a Fourier series according to
y(t) = 12 a0 +
(aj cos(jωt) + bj sin(jωt)) .
j
Such a solution living in the infinite dimensional function space L2 cannot be
directly computed from the periodic BVP (7.18). Instead one aims at computing some Fourier-Galerkin approximation ym out of the finite dimensional
subspace Um ⊂ L2 according to the finite Fourier series ansatz
m
ym (t) = 12 a0 +
(aj cos(jωt) + bj sin(jωt)) ,
j=1
where aj , bj ∈ Rn . Insertion into the approximate periodic BVP
ym
= f (ym ) ,
ym (T ) = ym (0)
will require the coefficients of the derivative defined via
ym
(t) =
m
aj cos(jωt) + bj sin(jωt)
j=1
with
aj = jωbj ,
bj = −jωaj
(7.24)
7.3 Periodic Orbit Computation
345
as well as those of the right side defined via
m
f (ym (t)) =
(αj cos(jωt) + βj sin(jωt)) .
(7.25)
j=1
Upon inserting the two expansions into the BVP (7.24), we arrive at the
system of N = 2m + 1 relations
jωbj = αj ,
−jωaj = βj ,
j = 0, . . . , m ,
j = 1, . . . , m .
From the theory of Fourier transforms we know that the right side coefficients
can be computed via
2
αj =
T
T
f (ym (t)) cos(jωt) dt ,
T
2
βj =
T
t=0
f (ym (t)) sin(jωt) dt . (7.26)
t=0
Obviously, this Galerkin approach involves a continuous Fourier transform
which inhibits the construction of an approximation scheme for ym .
Therefore, already in 1965, M. Urabe [188] suggested to replace the continuous Fourier transform by a discrete Fourier transform, i.e., by trigonometric
interpolation over equidistant nodes
tk = T
k
,
N
k = 0, 1, . . . , N .
Formally speaking, the integrals in (7.26) are then approximated by their
trapezoidal sums defined over the selected set of nodes. As a consequence, we
now substitute the Galerkin approximation ym by a Galerkin-Urabe approximation defined via the modified Fourier series expansion
m
ŷm (t) = 12 â0 +
âj cos(jωt) + b̂j sin(jωt)
(7.27)
j=1
and the corresponding expansion for its derivative ŷm
(t). Instead of (7.25)
we now have a modified expansion
m
f (ŷm (t)) =
α̂j cos(jωt) + β̂j sin(jωt)
j=1
and instead of the representation (7.26) we obtain the well-known trigonometric expressions
α̂j =
2
N
N−1
f (ŷm (tk )) cos(jωtk ) ,
k=0
β̂j =
2
N
N −1
f (ŷm (tk )) sin(jωtk ) .
k=0
(7.28)
346
7 ODE Boundary Value Problems
We again require the N = 2m + 1 relations
jω b̂j = α̂j ,
j = 0, . . . , m ,
−jωâj = β̂j ,
j = 1, . . . , m .
(7.29)
As before, we insert the expansion for ŷm into the formal representation
(7.28) so that the coefficients α̂j , β̂j drop out and we arrive at a system
of nN equations, in general nonlinear. Originally, Urabe had suggested this
approach, also called harmonic balance method, for nonautonomous periodic
BVPs where T (or ω, respectively) is given in the problem so that the nN
unknown coefficients (â0 , â1 , b̂1 , . . . , âm , b̂m ) can, in principle, be computed.
For autonomous periodic BVPs as treated here, system (7.29) turns out to
be underdetermined with nN + 1 unknowns (â0 , â1 , b̂1 , . . . , âm , b̂m , ω) to be
computed. We observe that in this kind of approximation local nonuniqueness shows up just as in the stated original problem: given a solution with
computed coefficients âj , b̂j , then any trajectory defined by the modified coefficients
ã0 = â0 ,
ãj = âj cos(jωτ ) + b̂j sin(jωτ ) ,
b̃j = b̂j cos(jωτ ) − âj sin(jωτ ) ,
is also a solution, shifted by τ . Therefore we may simply transfer the GaussNewton methods for single orbit computation or for orbit continuation (see
the preceding sections) to the nonlinear mapping as just defined.
Numerical realization of Gauss-Newton method. For ease of writing,
we here ignore the difference between ym and ŷm and skip all ‘hats’ in the
coefficients. Then the underdetermined system has the form
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
F (z) = ⎜
⎜
⎜
⎜
⎜
⎜
⎝
−1
2 N
f (ym (tk ))
N k=0
..
.
2
N
2
N
⎞
⎟
⎟
⎟
⎟
⎟
N
−1
⎟
f (ym (tk )) cos(jωtk ) + jωbj ⎟
⎟ = 0,
⎟
k=0
⎟
N−1
⎟
f (ym (tk )) sin(jωtk ) − jωaj ⎟
⎟
k=0
⎠
..
.
⎛
⎞
a0
⎜ .. ⎟
⎜ . ⎟
⎜
⎟
⎜ aj ⎟
⎜
⎟
z=⎜
⎟
⎜ bj ⎟
⎜ . ⎟
⎝ .. ⎠
ω
in terms of a mapping F : RnN +1 −→ RnN . Herein z additionally enters via
the expression (7.27) understood to be inserted for ym . For the corresponding
Jacobian (nN, nN + 1)-matrix we may write block columnwise
F (z) = (Fa0 (z), Fa1 (z), Fb1 (z), . . . , Fam (z), Fbm (z), Fω (z)) .
Assume we have already computed the Fourier series expansion of the (n, n)matrix fy (ym ), say with coefficients
7.3 Periodic Orbit Computation
Aj =
2
N
N −1
fy (ym (tk )) cos(jωtk ) ,
Bj =
k=0
2
N
347
N −1
fy (ym (tk )) sin(jωtk ) .
k=0
Then a straightforward calculation reveals that
⎛
0
..
.
⎜
⎜
⎜
⎜
Fω (z) = ⎜ jaj
⎜
⎜ −jbj
⎝
..
.
⎛
⎞
⎜
⎜
⎜
⎜
Fa0 (z) = ⎜
⎜
⎜
⎜
⎝
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎠
1
2 A0
⎞
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
1
⎟
2 Aj ⎠
..
.
..
.
1
B
2 j
and, for l = 1, . . . , m:
⎛
⎜
⎜
⎜
⎜
Fal (z) = ⎜
⎜
⎜
⎜
⎝
and
1
(Bl+j
2
− Bl−j ) + jωδjl In
1
2 (Al+j
⎛
⎜
⎜
⎜
⎜
Fbl (z) = ⎜
⎜
⎜
⎜
⎝
⎞
Al
..
.
+ Al−j )
..
.
⎞
Bl
..
.
1
(−BAl+j
2
1
(Bl+j
2
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎠
+ Al−j )
+ Bl−j ) − jωδjl In
..
.
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎠
In this representation, certain indices run out of the permitted index set:
whenever an index l > m appears, then replace Al by AN −l and Bl by
−BN −l ; whenever l < 0, then replace Al by A−l and Bl by −B−l . In this
way all computations can be performed using the FFT algorithm for f and fy ,
assuming that an iterate ym is at hand—which it is during the Gauss-Newton
iterations for single orbit computation or orbit continuation (see preceding
sections).
Adaptivity device. Up to now, we have not discussed the number m of
terms necessary to obtain an approximations to prescribed accuracy. Let | · |
denote the L2 [0, T ]-norm then we have
⎛
εm = |y − ym | = ⎝
∞
j=m+1
⎞1
2
a2j + b2j ⎠ .
348
7 ODE Boundary Value Problems
If we repeat the Galerkin-Urabe procedure with m replaced by M m then
we may choose the computationally available term
⎛
[εm ] = |yM − ym | = ⎝
M
2
⎞1
2
a2j + bj ⎠ ≤ εm
j=m+1
as a reasonable error estimate. Upon again ignoring the difference between
the Fourier coefficients ym and ŷm , we may apply a well-known approximation
result from Fourier analysis: assume that the unknown function y and all its
approximations ym are analytic, then the coefficients obey some exponential
decay law, which we write in the form
.
εm = Ce−γm ,
where the coefficients C, γ are unknown a-priori and need to be estimated.
For this purpose, let the optimal number m∗ be such that
.
εm∗ = TOL
and assume that this is not yet achieved for the actual index m. Then a
short calculation (see also Exercise 7.6 for more details) shows that m∗ can
be estimated by the adaptive rule (with some further index l m)
log([εm ]/ TOL)
.
m∗ = m + (m − l)
.
log([εl ]/[εm ])
(7.30)
Only with such an adaptivity device added, the Galerkin-Urabe (also: harmonic balance) method can be expected to supply reliable computational
results.
From (7.28) and (7.29) we may readily observe that in the Galerkin-Urabe
approach the BVP (7.24) has been tacitly replaced by the discrete boundary
value problem
ŷm
(tk ) = f (ŷm (tk )) ,
k = 0, 1, . . . , N ,
ŷm (T ) = ŷm (0) ,
wherein t0 = 0, tN = T by definition. The boundary conditions are implicitly
taken into account by the Fourier ansatz. One of the conditions, at t = 0 or
at t = T , can be dropped due to periodicity so that there are nN so-called
collocation conditions left. By construction, the method inherits the symmetry of the BVP with respect of an interchange of the boundaries t = 0 and
t = T —indicating that this computational approach treats periodic BVPs as
spacelike BVPs—as opposed to the preceding Sections 7.3.1 and 7.3.2 where
multiple shooting approaches for timelike BVPs have been discussed. The
following Section 7.4 is fully devoted to collocation methods for spacelike
BVPs—there, however, in connection with polynomial approximation.
7.4 Polynomial Collocation for Spacelike BVPs
349
7.4 Polynomial Collocation for Spacelike BVPs
In the collocation approach the interval [a, b] is subdivided into a partition
Δ = {a = t1 < t2 < · · · < tm = b},
m > 2,
where each subinterval Ij = [tj , tj+1 ] of length τj = tj+1 − tj is further
subdivided by s internal nodes, the so-called collocation points
tji = tj + ci τj , i = 1, . . . , s ,
0 ≤ c1 < · · · < cs ≤ 1
corresponding to some quadrature rule of order p with nodes ci . Let Δ∗ denote
the union of all collocation points—to be distinguished from the above defined
coarse mesh Δ.
Let u denote the collocation polynomial to be computed for the given BVP.
The collocation polynomial is defined via the n boundary conditions
Fm = r(u(a), u(b)) = 0
typically assumed to be linear separated (cf. [9, 71])
Fm = Au(a) + Bu(b) − d = 0 ,
(7.31)
so that all components arising therein can be fixed. At the ‘internal’ nodes
we require the (m − 1)sn ‘local’ collocation conditions
Fji = u (tji ) − f (u(tji )) = 0,
tji ∈ Δ∗
(7.32)
often in the scaling invariant form (i.e. invariant under rescaling of the variable t)
Fji = τj (u (tji ) − f (u(tji ))) = 0,
tji ∈ Δ∗ .
Finally, the (m − 1)n ‘global’ collocation conditions
s
Fj = u(tj+1 ) − u(tj ) − τj
bl f (u(tjl )) = 0 ,
tj ∈ Δ
(7.33)
l=1
must hold, wherein the quadrature rule implies the relation
s
bl = 1 .
l=1
By construction, the collocation approach is invariant under a ↔ b whenever
a symmetric quadrature rule with
ci = 1 − cs+1−i ,
bi = bs+1−i ,
i = 1, . . . , s
is selected. In fact, symmetric collocation methods are realized in nearly
all public domain codes, since they permit the highest possible convergence
orders p by one out of the following two options:
350
7 ODE Boundary Value Problems
• Gauss methods. Here collocation points are selected as the nodes of GaussLegendre quadrature. This leads to the highest possible order p = 2s. The
simplest case with s = 1 is just the implicit midpoint rule. Since c0 > 0
and cs < 1, the nodes of the coarse mesh are not collocation points, i.e.,
Δ∗ ∩ Δ = ∅, and hence only u ∈ C 0 [a, b] is obtained.
• Lobatto methods. Here the collocation points are selected as the nodes of
Lobatto quadrature. The attainable order is p = 2s − 2. The simplest case
is the implicit trapezoidal rule with p = s = 2. Since c0 = 0 and cs = 1,
the nodes of the coarse mesh are included in the set of collocation points,
i.e., Δ∗ ∩ Δ = Δ. The lower order (compared with the Gauss methods)
comes with better global smoothness, since here u ∈ C 1 [a, b].
In algorithmic implementations, Gauss methods are usually preferred due to
their more robust behavior in nonsmooth BVPs.
Bibliographical Note. Efficient collocation methods have been implemented in the classical code COLSYS of U.M. Ascher, J. Christiansen, and
R.D. Russell and its more recent variant COLNEW by G. Bader and U.M. Ascher
[16]. An adaptive Gauss-Newton continuation method (as described in Section 5.2) has been implemented in the code COLCON by G. Bader and P. Kunkel
[17]. An advanced residual based inexact Gauss-Newton continuation method
has been designed for collocation by A. Hohmann [120] and realized in the
rather robust research code COCON. Unfortunately, that line of development
has not continued toward a fully satisfactory general purpose collocation
code. We will resume the topic, in Section 7.4.2 below, in the frame of error
oriented inexact Newton methods. Recently, this concept has regained importance in a novel multilevel algorithm for optimal control problems based on
function space complementarity methods—a topic beyond the present scope,
for details see M. Weiser and P. Deuflhard [197].
7.4.1 Discrete versus continuous solutions
From multiple shooting techniques we are accustomed to the fact that, whenever the underlying BVP has a locally unique solution, the discrete system
also has a locally unique solution—just look up Lemma 7.1 and the interpretation thereafter. This need not be the case for global discretization methods.
Here additional ‘spurious’ discrete solutions may occur that have nothing
to do with the unique continuous BVP solution. In [71, Section 8.4.1] this
situation has been analyzed for the special method based on the implicit
trapezoidal rule, the simplest Lobatto method already mentioned above. In
what follows we want to give the associated analysis for the whole class of
collocation methods. We will mainly focus on Gauss collocation methods,
which are the ones actually realized in the most efficient collocation codes.
Our results do, however, also apply to other collocation schemes.
7.4 Polynomial Collocation for Spacelike BVPs
351
Throughout this section we assume—without proof—that the BVP has a
unique solution y ∈ C p+1 [a, b] and that the collocation polynomial u is globally continuous, i.e., u ∈ C 0 [a, b], but only piecewise sufficiently differentiable,
p+1
u ∈ C p+1 [tj , tj+1 ], which we denote by u ∈ CΔ
[a, b]. In what follows we
restrict our attention to Gauss methods, which means p = 2s. Consequently,
p+1
the discretization error (t) = u(t) − y(t), t ∈ [a, b] satisfies ∈ CΔ
[a, b].
In order to study its behavior, we introduce norms over the grids Δ, Δ∗ , for
example
||Δ = max (t)
t∈Δ
in terms of some vector norm · . Let
τ=
max
j=1,...,m−1
τj
denote the maximum mesh size on the coarse grid Δ. With these preparations we are now ready to state a convergence theorem for Gauss collocation
methods.
Theorem 7.2 Notation as just introduced. Consider a BVP on [a, b] with
linear separated boundary conditions and a right side f that is p-times differentiable with respect to its argument. Assume that the BVP has a unique
solution y ∈ C p+1 [a, b]. Let this solution be well-conditioned with bounded
interval condition number ρ̄. Define a global Lipschitz constant ω via
fy (v) − fy (w) ≤ ωv − w .
(7.34)
Consider a Gauss collocation scheme based on a quadrature rule of order
p+1
p = 2s. Let the discrete BVP have a collocation solution u ∈ CΔ
[a, b]
that is consistent with the BVP solution y. Let γ, γ ∗ denote error coefficients
corresponding to the Gauss quadrature rule and depending on the smoothness
of the right hand side f . Then, for
− 1
τ ≤ 2ωγ ∗(ρ̄|b − a|)2 s+1 ,
(7.35)
the following results hold:
(I) At the local (internal) nodes the pointwise approximation satisfies
|u − y |Δ∗ ≤ 2ρ̄|b − a|γ ∗ τ s+1 .
(II) At the global nodes superconvergence holds in the sense that
|u − y |Δ ≤ ρ̄|b − a| 2ω(ρ̄|b − a|γ ∗ τ )2 + γ τ 2s .
(7.36)
(7.37)
Proof. I. We begin with deriving a perturbed variational equation for the
p+1
discretization error ∈ CΔ
[a, b]. For t ∈ Ij = [tj , tj+1 ] we may write
352
7 ODE Boundary Value Problems
(t) − fy (y(t))(t) = δf (t) + δϕ(t) ,
where
(7.38)
δf (t) = u (t) − f (u(t))
and
δϕ(t)
=
f (u(t)) − f (y(t)) − fy (y(t))(t)
1
=
fy (y(t) + Θ(t)) − fy (y(t)) (t) dΘ .
Θ=0
The above first term δf vanishes at the collocation points and gives rise to
the local upper bound for the interpolation error (see, e.g., [71, Thm. 7.16])
max δf (σ) ≤ γτjs .
σ∈Ij
(7.39)
The second term δϕ contains the nonlinear contribution and satisfies the
pointwise estimate
δϕ(t) ≤ 12 ω(t)2 .
(7.40)
By variation of constants (see Exercise 7.7) the differential equation (7.38)
can be formally solved for t ∈ [tj , tj+1 ] to yield
t
(t) = W (t, tj )(tj ) +
W (t, σ) (δf (σ) + δϕ(σ)) dσ ,
σ=tj
where W (·, ·) denotes the (Wronskian) propagation matrix, the solution of the
unperturbed variational equation—see [71, Section 3.1.1]. This, however, is
just a representation of the solution of the IVP on each subinterval. Therefore,
any estimates based on this formula would bring in the IVP condition number,
which we want to avoid for spacelike BVPs.
For this reason, we need to include the boundary conditions, known to be
linear separated, so that
δr = A(t1 ) + B(tm ) = 0 .
Upon combining these results, we obtain the formal global representation
b
(t) =
G(t, σ) (δf (σ) + δϕ(σ)) dσ ,
σ=a
where G(·, ·) denotes the Green’s function of the (linear) variational BVP.
Its global upper bound is the condition number of the (nonlinear) BVP (as
defined in [71, Section 8.1.2]):
7.4 Polynomial Collocation for Spacelike BVPs
353
ρ̄ = max G(t, t̄) .
t,t̄∈[a,b]
Since the BVP is assumed to be well-conditioned, the above condition number
is bounded. Note that, by definition, the Green’s function has a jump at t = t̄
such that
G(t, t+ ) − G(t, t− ) = I .
For the subsequent derivation we decompose the integral into subintegrals
over each of the subintervals such that
m−1
tk+1
(t) =
G(t, σ) (δf (σ) + δϕ(σ)) dσ .
k=1 σ=t
k
II. We are now ready to derive upper bounds for the discretization error.
For tj ∈ Δ, we may directly apply the corresponding Gauss quadrature rule,
since the above jumps occur only at the boundaries of each of the subintegrals.
Along this line we obtain
m−1
(tj ) =
bl G(tj , tkl )δϕ(tkl ) + Γk (·)τk2s .
s
τk
k=1
l=1
The argument in the remainder term Γk (·) is dropped, since we are only
interested in its global upper bound, say
Γk (·) ≤ ρ̄γ .
Introducing pointwise norms on the coarse grid Δ, and exploiting (7.40) and
(7.36), we are then led to
(tj ) ≤ ρ̄|b − a| |δϕ|Δ∗ + γτ 2s ,
which yields
||Δ ≤ ρ̄|b − a|
1
2
2 ω||Δ∗
+ γτ 2s .
(7.41)
Next we consider arguments t = tji ∈ Δ∗ . In this case the jumps do occur
inside one of the subintegrals. Hence, we have to be more careful in our
estimate. We start with
(t) ≤
m−1
tk+1
G(t, σ) (δf (σ) + δϕ(σ)) dσ
k=1 σ=t
k
from which we immediately see that the integrals over Ik for k = j can be
treated as before
tk+1
σ=tk
G(t, σ) (δf (σ) + δϕ(σ)) dσ ≤ ρ̄τk γτk2s + 12 ω||2Δ∗ .
354
7 ODE Boundary Value Problems
For k = j, however, we just obtain
tj+1
G(t, σ) (δf (σ) + δϕ(σ)) dσ≤ ρ̄τj max δf (σ) + 12 ω||2Δ∗ .
σ∈Ij
σ=tj
If we recall (7.39) and define the quantity
γ ∗ τ = (γτ s + γ
τ
),
|b − a|
we end up with the quadratic inequality
||Δ∗ ≤ ρ̄|b − a|( 12 ω||2Δ∗ + γ ∗ τ s+1 ) .
(7.42)
For the solution of this inequality, we introduce the majorant ||Δ∗ ≤ ¯ generating the quadratic equation
¯ = ρ̄|b − a|( 12 ω¯
2 + γ ∗ τ s+1 ) .
For the discriminant to be nonnegative we need to require
1
,
2ωγ ∗(ρ̄|b − a|)2
τ s+1 ≤
which is just statement (7.35) of the theorem. This situation is represented
graphically in Figure 7.5.
¯
¯1
¯2
Fig. 7.5. Left and right side of quadratic inequality (7.42).
With the notations
α=
1
,
ρ̄|b − a|ω
− 1
τc = 2ωγ ∗ (ρ̄|b − a|)2 s+1
we obtain the two majorant roots
7.4 Polynomial Collocation for Spacelike BVPs
¯1 =
α(τ /τc )s+1
,
1 + 1 − (τ /τc )s+1
2 = α 1 +
¯
355
1 − (τ /τc )s+1 = 2α − ¯1 .
For τ ∈ [0, τc [ we obtain the bounds
0≤¯
1 ≤ α− ,
2α ≥ ¯2 ≥ α+ .
The situation is depicted in Figure 7.6.
¯
2α
2
¯
α
1
¯
τ
τc
Fig. 7.6. Error bounds for consistent discrete solution (¯
≤ ¯1 ) and spurious or
‘ghost’ solutions (¯
≥ ¯2 ).
First we pick ¯1 , the root consistent with the continuous BVP, and are led to
the approximation result
||Δ∗ ≤ ¯1 ≤ 2ρ̄|b − a|γ ∗ τ s+1 ,
which verifies statement (7.36). Second we study the root ¯2 . Again under
the meshsize constraint (7.35), the quadratic inequality can be seen to characterize a further discrete solution branch by
||Δ∗ ≥ ¯2 .
Obviously, the proof permits the existence of inconsistent discrete solutions.
However, if such solutions exist, they are well-separated from the consistent
ones as long as τ < τc .
As a final step of the proof, we may just insert the upper bound (7.36) into
(7.41) and arrive at
||Δ ≤ ρ̄|b − a| 12 ω(ρ̄|b − a|γ ∗ τ s+1 )2 + γτ 2s
≤ ρ̄|b − a| 12 ω(ρ̄|b − a|γ ∗ τ )2 + γ τ 2s .
This is the desired superconvergence result (7.37) and thus completes the
proof.
356
7 ODE Boundary Value Problems
Of course, the theorem does not state anything about the situation when the
meshsize restriction (7.35) does not hold, i.e. for τ > τc .
The above discussed possible occurrence of ‘ghost’ solutions, i.e. of inconsistent discrete solutions, which have nothing to do with the continuous solution, has been experienced by computational scientists, both in ODEs and
in PDEs. In words, the above theorem states that a computed collocation
solution u is a valid approximation of the BVP solution y only, if the applied
mesh is ‘sufficiently fine’ and if this solution ‘essentially is preserved’ on successively finer meshes. This means that the actual uniqueness structure of a
collocation solution cannot be revealed by solving just one finite-dimensional
problem. Rather, successive mesh refinement is additionally needed as an algorithmic device to decide about uniqueness. This paves the way to Newton
methods in function space, also called quasilinearization—to be treated in
the next section.
7.4.2 Quasilinearization as inexact Newton method
Instead of a Newton method for the discrete nonlinear system (7.31), (7.32),
and (7.33), the popular collocation codes realize some quasilinearization technique, i.e., a Newton method in function space. Of course, approximation
errors are unavoidable, which is why inexact Newton methods in function
space are the correct conceptual frame.
We start with the exact ordinary Newton iteration in function space
yk+1 (t) = yk (t) + δyk (t) ,
k = 0, 1, . . . .
Herein the Newton corrections δyk satisfy the linearized BVP, which is the
perturbed variational equation with linear separated boundary conditions:
δyk − fy (yk (t))δyk
= − (yk (t) − f (yk (t))) ,
t ∈ [a, b] ,
Aδyk (a) + Bδyk (b) = 0 .
Newton’s method in the infinite dimensional function space can rarely be
realized, apart from toy problems. Instead we study here the corresponding
exact Newton method in finite dimensional space, i.e., the space spanned by
the collocation polynomials on the grids Δ and Δ∗ (as introduced in the
preceding section). Formally, this method replaces y, δy by their polynomial
representations u, δu, where
δuk (tji ) − fy (uk (tji ))δuk (tji ) = −δfk (tji ) = − (uk (tji ) − f (uk (tji ))) ,
Aδuk (a) + Bδuk (b) =
0.
(7.43)
If we include a damping factor λk to expand the local domain of convergence
of the ordinary Newton method, we arrive at
7.4 Polynomial Collocation for Spacelike BVPs
uk+1 (t) = uk (t) + λk δuk (t) .
357
(7.44)
Note that here Theorem 7.2 applies with global Lipschitz constant ω = 0. As a
consequence there are no spurious solutions δu to be expected—which clearly
justifies the use of quasilinearization. Nevertheless, the chosen mesh might
be not ‘fine enough’ to represent the solution correctly—see the discussion
on mesh selection below.
Linear band system. Within each iteration, dropping the index k, we
have to solve the finite-dimensional linear system consisting of the (scaling
invariant) local collocation conditions
Fji = τj (δu (tji ) − fy (u(tji ))δu(tji ) + δf (tji )) = 0 ,
the global collocation conditions
s
Fj = δu(tj+1 ) − δu(tj ) − τj
bl (fy (u(tjl ))δu(tjl ) − δf (tjl )) = 0
(7.45)
l=1
and the boundary conditions
Fm = Aδx1 + Bδxm = 0 .
In most implementations, the local collocation conditions are realized in the
equivalent initial value problem form
s
Fji = δu(tji ) − δu(tj ) − τj
ail (fy (u(tjl ))δu(tjl ) − δf (tji )) .
(7.46)
l=1
In this case, the local variables u(tj1 ), . . . , u(tjs ) can be condensed—which
means expressed in terms of the global variables u(tj ). However, unlike the
equations (7.45) and (7.43), the part (7.46) is not symmetric with respect
to a ↔ b. The separated boundary conditions can be dropped by fixing the
proper components of the boundary values.
The remaining linear system has block tridiagonal structure. It is usually
solved by some global direct elimination method such as the modified band
solver due to J.M. Varah [191].
In the already mentioned Gauss-Newton continuation method for parameter dependent BVPs (cf. [17]), the discretized linear system is just enhanced
by a further column, which requires a slight modification of the elimination method; the corresponding rank-deficient Moore-Penrose pseudoinverse
is then simply realized via the representation (5.29) as presented in Section
5.2.
358
7 ODE Boundary Value Problems
Norms. As in Section 7.4.1 above we here also write · for a local vector
norm. With | · | we will mean the canonical C 0 -norm defined as
||0,[a,b] = max (t) ,
t∈[a,b]
whose discretization is
||0,Δ = max (t) .
t∈Δ
If we scale the L2 -norm appropriately, we find that
⎛
||2,[a,b] = ⎝
1
b−a
⎞ 12
b
(t)2 dt⎠ ≤ ||0,[a,b] .
a
From this, we may turn to the corresponding discretization
⎛
||2,Δ = ⎝
1
b−a
m−1
s
τj
j=1
⎞1
2
bl (tjl )2 ⎠
l=1
and obtain the corresponding relation
||2,Δ ≤ ||0,Δ∗ .
Note that, due to Gaussian quadrature, we have the approximation property
||2,Δ = ||2,[a,b] + O(τ 2s ) ≤ ||0,[a,b] + O(τ 2s ) .
Below we will not distinguish between any of these essentially equivalent
norms and write subscripts only where necessary.
Discretization error estimates. The above exact finite dimensional Newton method can also be viewed as an inexact Newton method in function
space, if we include the discretization errors into a unified mathematical
frame. In fact, any adaptive collocation method will need to control the arising discretization error
(t) = δu(t) − δy(t)
at least via some estimate of it, in some suitable norm (see above). For convenience, we again use the notation Ij = [tj , tj+1 ] for the subintervals of the
coarse mesh Δ.
Before we start, let us draw a useful consequence of Theorem 7.2: we interpret the Gaussian collocation method as an implicit Runge-Kutta method
(compare, e.g., [115, 71]) and thus immediately obtain (for k < s)
δu(k) (t)−δy (k) (t) = δy (s+1) (tj )τjs+1 Ps(k) (
t − tj
)(1+O(τj ))+O(τj2s ) , (7.47)
τj
7.4 Polynomial Collocation for Spacelike BVPs
359
where P is a polynomial of degree s—see, e.g., [9, Section 9.3]. Since the
local collocation polynomials are of order s, their derivative δu(s) is piecewise
constant, i.e.,
δu(s) (t) = constj , ,
δu(s+1) (t) = 0 ,
t ∈ Ij .
At first glance, we do not seem to have a reasonable approximation of δy (s+1)
that could be used to estimate the error bound (7.47). However, we may
overcome this lack of information by defining piecewise linear functions δzs
via the interpolation conditions at the subinterval midpoints t̄j = tj + 12 τj
such that
δzs (t̄j ) = δu(s) (t̄j ) .
The situation is depicted schematically in Figure 7.7. The derivative δzs is
piecewise constant with jumps at the subinterval midpoints t̄j .
t̄1
t̄2
t̄3
Fig. 7.7. Piecewise linear approximation δzs of piecewise constant function δu(s)
(s)
Since Ps ( 12 ) = 0, (7.47) implies the superconvergence result
δu(s) (t̄j ) − δy (s) (t̄j ) = O(τj2s ) ,
from which we obtain the approximation property
δzs (t) = δy (s+1) (tj ) + O(τj ) ,
t ∈ Ij .
Hence, in first order of the local mesh size τj , we obtain the componentwise
estimate
.
|δu(t) − δy(t)| = |δzs (tj )|τjs+1 , t ∈ Ij .
If we average according to the rule
|δzs (t̄1 )| = |δzs (t1 )| ,
|δzs (t̄j )| =
1
2
|δzs (t̄m )| = |δzs (tm )| ,
(|δzs (tj )| + |δzs (tj+1 )|) ,
j = 2, . . . , m − 1 ,
we arrive at the cheaply computable local discretization error estimates
.
max |δu(t) − δy(t)| = j = |δzs (t̄j )|τjs+1 .
(7.48)
t∈Ij
In [120], A. Hohmann has suggested a realization with variable order pj = 2sj
in different subintervals Ij ; for such an h − p-strategy, the above estimation
360
7 ODE Boundary Value Problems
technique will no longer be applicable. Instead, the Gauss interpolation polynomial is extended by the additional boundary nodes tj , tj+1 . This kind of
collocation polynomial is of order sj + 2. Its difference to the computed solution can then serve as an estimate of order sj + 1 replacing (7.48).
Global mesh selection. This issue is crucial in every global discretization
method, at least when the numerical solution of really challenging spacelike
BVPs is envisioned, including the possible occurrence of internal or boundary
layers. Typically, a starting mesh Δ will be defined at the beginning of the
discretization process, which may already contain some information about
the expected behavior of the solution. Within each quasilinearization step
the componentwise global discretization error estimate
|| =
max
j=1,..,m−1
j
can be computed as indicated above. In order to minimize this maximum
subject to the constraint
m−1
τj = b − a ,
j=1
the well-known greedy algorithm leads to the requirement of equidistribution
of the local discretization errors. Therefore, any reasonable mesh selection
device will aim, at least asymptotically, at
j = Cj τjs+1 ≈ const ,
j = 1, . . . , m − 1
with coefficients Cj as defined above; an equivalent formulation is
j ≈ =
1
m−1
m−1
l .
l=1
There are various options to realize this equidistribution principle. In the
code family COLSYS the number m of nodes is adapted such that the global
error estimate eventually satisfies
|| ≤ TOL
in terms of the user prescribed error tolerance TOL. Some codes also permit
nonnested successive meshes.
In view of the theoretical approximation results (as given, e.g., in the previous
section), COLSYS uses a further global mesh refinement criterion based on an
affine covariant Newton method including a damping strategy [60, 63, 9].
Whenever the damping factor turns out to be ‘too small’, i.e., whenever
λk < λmin occurs, for some prescribed threshold value λmin 1, then a new
mesh with precisely halved local stepsizes is generated. Note that in Newton
7.4 Polynomial Collocation for Spacelike BVPs
361
methods, which are not affine covariant, such a criterion may be activated
in the wrong situation: not caused by the nonlinearity of the problem, but
by the ill-conditioning of the discrete equations—which automatically comes
with the fact that the BVP operator is noncompact.
Local mesh refinement. Instead of the mesh selection devices realized
within COLSYS we here want to work out an alternative option in the spirit
of adaptive multilevel methods for partial differential equations (to be treated
in Section 8.3 below). Let
Δ0 ⊂ Δ1 ⊂ · · · ⊂ Δd
denote a sequence of nested meshes. By construction, the mesh Δ0 is just
the given initial mesh, while the mesh Δ1 is obtained by halving of all subintervals. All further meshes can be obtained using an adaptivity device based
on local extrapolation—a technique that has been suggested by I. Babuška
and W.C. Rheinboldt [12] already in 1978, there for finite element methods
in partial differential equations. For details we here recur to Section 9.7.1 of
the elementary textbook [77], where this technique is explained in the simple
context of numerical quadrature. Assume we have already computed local error estimates on two consecutive meshes, on the given mesh Δ and its coarser
predecessor Δ− . Let I := (tl , tm , tr ) ∈ Δ− denote an interval bisected into
subintervals Il , Ir ∈ Δ, where
Il := tl , 12 (tl + tm ), tm and Ir := tm , 12 (tr + tm ), tr .
Repeated refinement leads to a recursive binary tree—see Figure 7.8.
I
r
r
+
Il
b
r
b
Ill
r
J
J
b
Q
r
Q
Q
Ir
b
^
J
Ilr
r
r
b
Q
s
Irl
r
J
J
b
b
^
J
Irr
r
b
Fig. 7.8. Double refinement of subinterval I := (tl , tm , tr )
In contrast to our above approximation results, we make the following more
general assumption for the local discretization error
362
7 ODE Boundary Value Problems
.
(Ij ) = Cτjγ
(7.49)
with a local order γ and a local problem dependent constant C to be roughly
identified; note that above we derived γ = s in the upper bounds, which
may be unrealistic in a specific example. Let I be a subinterval obtained by
refinement; then we denote the starting interval from the previous mesh by
I − , i.e., Ir− = Il− = I. Dropping the local index j, assumption (7.49) then
implies
.
.
(I − ) = C(2τ )γ = 2γ Cτ γ = 2γ (I) ,
from which we conclude that
.
.
(Il ) = Cτ γ 2−γ = (I)(I)/(I − ) .
Thus, through local extrapolation, we have obtained a local error prediction
+ (I) :=
2 (I)
≈ ε(Il ) .
(I − )
(I)
κ(Δ)
+ (I)
a
b
Fig. 7.9. Error distributions: before refinement: (I), prediction after global
refinement: + (I), prediction after adaptive refinement: bold line.
We can therefore estimate in advance, what effect a refinement of an interval
I ∈ Δ would have. We only have to fix a threshold value for the local errors,
above which we refine an interval. In order to do this, we take the maximal
local error, which we would obtain from a global refinement , i.e., refinement
of all subintervals I ∈ Δ, and define
κ(Δ) := max + (I) .
I∈Δ
In order to illustrate the situation, we plot the computed estimated errors
(I) together with the predicted errors + (I) in a smoothed histogram, see
Figure 7.9.
7.4 Polynomial Collocation for Spacelike BVPs
363
Following the aim of local error equidistribution, we do not need to refine
near the left and right boundary; rather, refinement will pay off only in the
center region. We thus arrive at the following refinement rule: Refine only
those intervals I ∈ Δ, for which
(I) ≥ κ(Δ) .
This yields the error distribution displayed in bold line in Figure 7.9, which
is narrower than both the original distribution and the one to be expected
from global refinement.
Clearly, repeated refinement will ultimately generate some rough error equidistribution, provided the local error estimation technique is reliable, which here
means that the finest meshes need to be ‘fine enough’ to activate the superconvergence properties of Gauss collocation methods.
Error matching. If we view quasilinearization as an inexact Newton
method in function space, we need to match the Newton corrections δuk and
the discretization errors δuk − δyk . In his dissertation [120], A. Hohmann
worked out a residual based inexact Newton method using a Fredholm basis
for the local representation of the collocation polynomials and obtained some
rather robust algorithm. Here, however, we want to realize the error oriented
local (Section 2.1.5) and global Newton methods (Section 3.3.4) within the
collocation code COLSYS—transferring the finite dimensional case therein to
the present infinite dimensional one.
At each iteration index k, identify uk = yk —as a common starting point,
say—and define the two Newton corrections in function space
F (uk )δuk = −F (uk ) + rk ,
F (uk )δyk = −F (uk ) .
(7.50)
In contrast to the setting in finite dimensional inexact Newton methods (see
Chapters 2 and 3), here the residual rk is not generated by some inner iteration, but by the discretization error such that
F (uk )(δuk − δyk ) = rk .
In collocation, the situation is characterized by the fact that we have cheap
computational estimates of the relative discretization error
δk =
|δuk − δyk |
|δuk |
available in some (approximate) norm | · |—compare (3.50). This quantity
essentially depends on the selected mesh. For successively fine meshes, this
quantity will approach zero, which is part of the asymptotic mesh independence to be discussed in detail in Section 8.1 below (see also Exercise 8.3).
The iteration (7.44) is realized as a finite dimensional Newton iteration with
adaptive trust region (or damping) strategy—as long as the mesh is kept
364
7 ODE Boundary Value Problems
unchanged. However, in order to really solve the BVP and not just some discrete substitute system with unclear approximation quality, the mesh should
be adapted along with the iteration: the idea advocated here is to aim at
some asymptotic confluence with an exact Newton iteration based on the
correction δyk . In view of (3.55) and the analysis in Section 3.3.4, we will
require that, for some ρ ≤ 1,
δk ≤
ρ
≤
2(1 + ρ)
1
4
for λk < 1 .
(7.51)
As soon as the iteration swivels in the ordinary Newton phase with λ = 1
throughout, then a more careful consideration is needed, which can be based
on the following theoretical estimates.
Theorem 7.3 Let δuk and δyk denote the inexact and exact Newton corrections as defined in (7.50). Let ω be the affine covariant Lipschitz constant
defined via
|F (u)−1 (F (v) − F (w)) | ≤ ω|v − w| .
Then, with the notation of the present section, we obtain
(I) for the ordinary exact Newton method in finite dimension the quadratic
convergence result
|δuk+1 | ≤ 12 hδk |δuk |
with hδk = ω|δuk |,
(II) for the stepwise ‘parallel’ ordinary inexact Newton method in function
space, as defined in (7.50), the mixed convergence results
|δyk+1 | ≤
1 δ
2 hk
+ (1 + hδk )δk
|δyk |
1 − δk
and, under the additional matching assumption for some safety factor
ρ ≤ 1,
hδk
δk ≤ 12 ρ
,
(7.52)
1 + hδk
the modified quadratic convergence results
|δyk+1 | ≤ 12 (1 + ρ)
where contraction is realized, if hδ0 <
hδk
|δyk | ,
1 − δk
2 (1 − δ0 )
.
1+ρ
Proof. Part (I) of the theorem is standard. Part (II) is a slight modification
of Theorem 2.11 in Section 2.1.5, there derived for the inner iterative solver
GBIT, which does not satisfy any orthogonality properties. The definition of
7.4 Polynomial Collocation for Spacelike BVPs
365
hδk is different here; it can be directly inserted into the intermediate result
(2.54).
Clearly, we will select the same value of ρ in both (7.51) for the damped
Newton method and (7.52) for the ordinary Newton method.
Recall that the derivation of (7.51) required that λk needs to be chosen according to an adaptive trust region strategy based on computationally available Kantorovich estimates [hδk ] ≤ hδk . We are still left with the construction
of such estimates. Since the right hand upper bound in (7.52) is a monotone increasing function of hδk , we may then construct an adaptive matching
strategy just replacing hδk by its lower bound [hδk ]. From the above lemma we
directly obtain the a-posteriori estimates
[hδk ]1 = 2Θk ≤ hδk , where Θk =
|δuk+1 |
|δuk |
and the a-priori estimate
2
[hδk ] = 2Θk−1
≤ hδk .
With these preparations we are led to the following informal
Error matching algorithm. As long as the finite dimensional global Newton method is still damped, we realize δk ≤ 1/4 via appropriate mesh selection
as given above. Let the index k = 0 characterize the beginning of the local
Newton method with λ0 = 1. Then the following steps are required (skipping
emergency exits to avoid infinite loops):
1. k = 0: Given u0 , compute δu0 and its norm |δu0 |. Compute the discretization error estimate δ0 , e.g., via the suggestion (7.48).
2. If δ0 > 14 , then refine the mesh and goto 1,
else u1 = u0 + δu0 .
3. k ≥ 1: Given uk , compute δuk , its norm |δuk | and the contraction factor
Θk−1 =
|δuk |
.
|δuk−1 |
If Θk−1 > 1, then realize an adaptive trust region strategy with λ < 1 as
described in Section 3.3.4,
else compute the discretization error estimate δk .
4. If
2
ρΘk−1
ρ
δk > min
,
,
2
2(1 + ρ) 1 + Θk−1
then refine the mesh and goto 3,
else uk+1 = uk + δuk → uk and goto 3.
366
7 ODE Boundary Value Problems
If the adaptive collocation algorithm is performed including the matching
strategy as described here, the amount of work will clearly increase from step
to step, since the required discretization error estimate will ask for finer and
finer (adaptive) meshes. Of course, the process may be modified such that at
some iterate the value of δk is frozen—with the effect of eventually obtaining
unreasonably accurate results uk .
Remark 7.2 An adaptive collocation method as sketched here has not been
implemented so far. However, this kind of ideas has entered into the adaptive
multilevel collocation method used within a function space complementarity
approach to constrained optimal control problems that has recently been
suggested and worked out by M. Weiser and P. Deuflhard [197].
Exercises
Exercise 7.1 Given the extended mapping (7.12), derive the expressions
(7.14) for the associated Newton corrections. Sketch details of the corresponding adaptive trust region method. In which way is a control of the actual parameter stepsize performed? Why should the initial iterate τ 0 satisfy
h(τ 0 ) = 0?
Exercise 7.2 In order to compute a periodic orbit, one may apply a gradient method [156] as an alternative to the Gauss-Newton method described
in Section 7.3. Let, in a single shooting approach, the functional
ϕ := 12 r22
be minimized. Then
grad ϕ = [E(0), f (y(T ))]T r
in the autonomous case. Show that
E T r = u(0) − u(T )
for some u satisfying
u = −fy (y)T u ,
u(T ) = r .
Discuss the computational consequences of this relation. Why is, nevertheless,
such a gradient method unsatisfactory?
Exercise 7.3 In multiple shooting techniques, assume that local rank reduction leads to a replacement of the Jacobian inverse J −1 by the generalized
inverse J − defined in (7.5).
Exercises
367
a) Verify the four axioms (7.6) using the Penrose axioms for the MoorePenrose pseudo-inverse.
b) In the notation of Section 4.3.5, let λk ∼ [ωk ]−1 denote a damping factor estimate associated with the Jacobian inverse Jk−1 and λk ∼ [ωk ]−1
the analog estimate associated with the generalized inverse Jk− . Give a
computationally economic expression for λk . Show that, in general, the
relation λk > λk need not hold, unless m = 2.
Exercise 7.4 Consider a local rank reduction in multiple shooting techniques as defined by J − in (7.5) and the axioms (7.6). Show that for the
Gauss-Newton direction
Δx = −J − F .
the residual level function
m−1
T (x|I) := r22 +
Fj 22
j=1
is, in general, no longer an appropriate descent function in the sense of Lemma
3.11, whereas both the natural level function T (x|J − ) and the hybrid level
function T (x|R−1 J − ) still are.
Exercise 7.5
Consider a singular perturbation problem of the type
εy + f (t)y + g(t)y = h(t) ,
y(0) = y0 ,
y(T ) = yT
having one internal layer at some τ ∈]0, T [ with f (τ ) = 0. Assume that the
initial value problem is well-conditioned from τ to 0 and from τ to T so that
numerical integration in these directions can be conveniently performed. Consider a multiple shooting approach with m = 3 nodes {0, τ, T } that involves
numerical integration in the well-conditioned directions. Study the associated
Newton method in detail with respect to necessary Jacobian approximations,
condensed linear system solver, and iterative refinement sweeps. Discuss extensions for m > 3, where the above 3 nodes are among the selected multiple
shooting nodes. Interpret this approach in the light of Lemma 7.1.
Exercise 7.6 Consider the Fourier collocation method (also: Urabe or
harmonic balance method) as presented in Section 7.3.3.
a) Given the asymptotic decay law
.
εm = Ce−γm ,
verify the computationally available estimate (7.30) for the optimal number m∗ of terms needed in the Fourier series expansion (7.24). How many
368
7 ODE Boundary Value Problems
different Galerkin-Urabe approximations are at least required for this
estimate?
b) In the derivation of Section 7.3.3 we ignored the difference between the
Galerkin approximation ym with Fourier coefficients aj , bj and the acm
tually computed Galerkin-Urabe approximation with coefficients am
j , bj
depending on the truncation index m. Modify the error estimates so that
this feature is taken into account.
Exercise 7.7
Given a perturbed variational equation in the form
(t) − fy (y(t))(t) = δf (t) ,
t ∈ [a, b] ,
prove the closed analytic expression
t
(t) = W (t, a)(a) +
W (t, σ)δf (t)dσ .
σ=a
Hint: Apply the variation of constants method recalling that the propagation
matrix W (t, a) is a solution of the unperturbed variational equation.
8 PDE Boundary Value Problems
This chapter deals with Newton methods for boundary value problems
(BVPs) in nonlinear partial differential equations (PDEs). There are two
principal approaches: (a) finite dimensional Newton methods applied to given
systems of already discretized PDEs, also called discrete Newton methods,
and (b) function space oriented inexact Newton methods directly applied to
continuous PDEs, at best in the form of inexact Newton multilevel methods.
Before we discuss the two principal approaches in detail, we study the underlying feature of asymptotic mesh independence that connects the finite
dimensional and the infinite dimensional Newton methods, see Section 8.1.
In Section 8.2, we assume the standard situation in industrial technology
software, where the grid generation module is strictly separated from the
solution module. Consequently, nonlinear PDEs there arise as discrete systems of nonlinear equations with fixed finite, but usually high dimension n
and large sparse ill-conditioned Jacobian (n, n)-matrix. This is the domain of
applicability of finite dimensional inexact Newton methods. More advanced,
but typically less convenient in a general industrial environment, are function space oriented inexact Newton methods, which additionally include the
adaptive manipulation of discretization meshes within a multilevel or multigrid solution process. This situation is treated in Section 8.3 and compared
there with finite dimensional inexact Newton techniques.
We will not treat ‘multilevel Newton methods’ here (often also called ‘Newton
multilevel methods’), which are in between discrete Newton methods and
inexact Newton methods in function space; they have been extensively treated
in the classical textbook [113] by W. Hackbusch or in the synoptic study [135]
by R. Kornhuber, who uses an affine conjugate Lipschitz condition.
8.1 Asymptotic Mesh Independence
The term ‘mesh independence’ characterizes the observation that finite dimensional Newton methods, when applied to a nonlinear PDE on successively
finer discretizations with comparable initial guesses, show roughly the same
convergence behavior on all sufficiently fine discretizations. In this section, we
want to analyze this experimental evidence from an abstract point of view.
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance
and Adaptive Algorithms, Springer Series in Computational Mathematics 35,
DOI 10.1007/978-3-642-23899-4_8, © Springer-Verlag Berlin Heidelberg 2011
369
370
8 PDE Boundary Value Problems
Let a general nonlinear operator equation be denoted by
F (x) = 0 ,
(8.1)
where F : D → Y is defined on a convex domain D ⊂ X of a Banach space
X with values in a Banach space Y . We assume the existence of a unique
solution x∗ of this operator equation. The corresponding ordinary Newton
method in Banach space may then be written as
F (xk )Δxk = −F (xk ) ,
xk+1 = xk + Δxk ,
k = 0, 1, . . . .
(8.2)
In each Newton step, the linearized operator equation must be solved, which
is why this approach is often also called quasilinearization. We assume that an
affine covariant version of the classical Newton-Mysovskikh theorem holds—
like Theorem 2.2 for the finite dimensional case. Let ω denote the affine covariant Lipschitz constant characterizing the mapping F . Then the quadratic
convergence of Newton’s method is governed by the relation
|xk+1 − xk | ≤ 12 ω|xk − xk−1 |2 ,
where | · | is a norm in the domain space X.
In actual computation, we can only solve discretized nonlinear equations of
finite dimension, at best on a sequence of successively finer mesh levels, say
Fj (xj ) = 0 ,
j = 0, 1, . . . ,
where Fj : Dj → Yj denotes a nonlinear mapping defined on a convex domain Dj ⊂ Xj of a finite-dimensional subspace Xj ⊂ X with values in a
finite dimensional subspace Yj . The corresponding finite dimensional ordinary Newton method reads
Fj (xkj )Δxkj = −Fj (xkk ) ,
xk+1
= xkj + Δxkj ,
j
k = 0, 1, . . . .
In each Newton step, a system of linear equations must be solved, which may
be a quite challenging task of its own in discretized PDEs. The above Newton system can be interpreted as a discretization of the linearized operator
equation (8.2) and, at the same time, as a linearization of the discretization
of the nonlinear operator equation (8.1). Again we assume that Theorem 2.2
holds, this time for the finite dimensional mapping Fj . Let ωj denote the
corresponding affine covariant Lipschitz constant. Then the quadratic convergence of this Newton method is governed by the relation
xk+1
− xkj ≤ 12 ωj xkj − xk−1
2 ,
j
j
where · is a norm in the finite dimensional space Xj .
(8.3)
8.1 Asymptotic Mesh Independence
371
Under the assumptions of Theorem 2.2 there exist unique discrete solutions
x∗j on each level j. Of course, we want to choose appropriate discretization
schemes such that
lim x∗j = x∗ .
j→∞
From the synopsis of discrete and continuous Newton method, we immediately see that any comparison of the convergence behavior on different
discretization levels j will direct us toward a comparison of the affine covariant Lipschitz constants ωj . Of particular interest is the connection with the
Lipschitz constant ω of the underlying operator equation.
Consistent norms. An important issue for any comparison of affine covariant Lipschitz constants ωj on different discretization levels j is the choice of
consistent norms. In the mathematical treatment of Galerkin methods, we
will identify the norm | · | in X with the norm · in Xj ⊂ X. Moreover,
the needs of algorithmic adaptivity strongly advise to choose smooth norms.
These considerations bring us to Sobolev H p -norms to be properly selected
in each particular problem.
For non-Galerkin methods such as finite difference methods, the easiest way
to construct consistent norms is to discretize the function space norm | · |
appropriately, which directs us toward discrete H p -norms. For example, in
one-dimensional BVPs we may naturally use discrete L2 -norms (7.9) to treat
highly nonuniform meshes—see also their application in (7.16). For uniform
one-dimensional meshes, the discrete L2 -norms on level j differ from the
√
Euclidean vector norms in Rnj only by some dividing factor nj . Insertion of
the discrete L2 -norm instead of the Euclidean vector norm into the Lipschitz
condition (8.3) shows that this same factor would now multiply ωj . As long
as merely a single finite dimensional system were to be analyzed, this change
would not make a substantial difference, but only affect the interpretation.
A synoptic analysis of a sequence of nonlinear mappings, however, will be
reasonable only, if consistent discrete norms are used.
In what follows we will consider the phenomenon of mesh independence of
Newton’s method along two lines. First, we will show that the discrete Newton sequence tracks the continuous Newton sequence closely, with a maximal
distance bounded in terms of the mesh size; both of the Newton sequences
behave nearly identically until, eventually, a small neighborhood of the solution is reached. Second, we prove the existence of affine covariant Lipschitz
constants ωj for the discretized problems, which approach the Lipschitz constant ω of the continuous problem in the limit j −→ ∞; again, the distance
can be bounded in terms of the mesh size. Upon combining these two lines,
we finally establish the existence of locally unique discrete solutions x∗j in a
vicinity of the continuous solution x∗ .
To begin with, we prove the following nonlinear perturbation lemma.
372
8 PDE Boundary Value Problems
Lemma 8.1 Consider two Newton sequences {xk }, {y k } starting at initial
guesses x0 , y 0 and continuing as
xk+1 = xk + Δxk ,
y k+1 = y k + Δy k ,
where Δxk , Δy k are the corresponding ordinary Newton corrections. Assume
that an affine covariant Lipschitz condition with Lipschitz constant ω holds.
Then the following propagation result holds:
1 k
xk+1 − y k+1 ≤ ω
x − y k + Δxk xk − y k .
(8.4)
2
Proof. Dropping the iteration index k we start with
x + Δx − y − Δy
= x − F (x)−1 F (x) − y + F (y)−1 F (y)
= x − F (x)−1 F (x) + F (x)−1 F (y) − F (x)−1 F (y) − y + F (y)−1 F (y)
= x − y − F (x)−1 (F (x) − F (y)) + F (x)−1 (F (y) − F (x))F (y)−1 F (y)
−1
= F (x)
1
F (x)(x − y) −
F (y + t(x − y))(x − y) dt
t=0
+ F (x)−1 (F (y) − F (x))Δy.
Upon using the affine covariant Lipschitz condition, we arrive at
1
x
k+1
−y
k+1
≤
F (xk )−1 F (xk ) − F (y k + t(xk − y k )) (xk − y k ) dt
t=0
+ F (xk )−1 (F (y k ) − F (xk ))Δy k ω
≤ xk − y k 2 + ωxk − y k Δy k ,
2
which confirms (8.4).
With the above auxiliary result, we are now ready to study the relative behavior of discrete versus continuous Newton sequences.
A
Theorem 8.2 Notation as introduced. Let x0 ∈ Xj ⊂ X denote a given
starting value such that the assumptions of Theorem 2.2 hold including
h0 = ωΔx0 < 2 .
For the discrete mappings Fj and all arguments xj ∈ S(x0 , ρ +
2
)∩Xj define
ω
8.1 Asymptotic Mesh Independence
Fj (xj )Δxj = −Fj (xj ) ,
373
F (xj )Δx = −F (xj ) .
Assume that the discretization is fine enough such that
Δxj − Δx ≤ δj ≤
1
.
2ω
(8.5)
Then the following cases occur:
I. If h0 ≤ 1 −
1 − 2ωδj , then
1
,
ω
xkj − xk < 2δj ≤
II. If 1 −
1 − 2ωδj < h0 ≤ 1 +
xkj − xk ≤
1
(1 +
ω
k = 0, 1, . . . .
1 − 2ωδj , then
1 − 2ωδj ) <
2
,
ω
k = 0, 1, . . . .
In both cases I and II, the asymptotic result
lim sup xkj − xk ≤
k→∞
1
(1 −
ω
1 − 2ωδj ) < 2δj ≤
1
ω
can be shown to hold.
Proof. In [114, pp. 99, 160], E. Hairer and G. Wanner introduced ‘Lady
Windermere’s fan’ as a tool to prove discretization error results for evolution
problems based on some linear perturbation lemma. We may copy this idea
and exploit our nonlinear perturbation Lemma 8.1 in the present case. The
situation is represented graphically in Figure 8.1.
x3,0
x2,0
x1,0
δj
x0j = x0,0
x1j = x1,1
x∗
x2,1
x3,2
δj
δj
x2j = x2,2
x3j = x3,3
Fig. 8.1. Lady Windermere’s fan: continuous versus discrete Newton iterates.
The discrete Newton sequence starting at the given initial point x0j = x0,0
is written as {xk,k }. The continuous Newton sequence, written as {xk,0 },
374
8 PDE Boundary Value Problems
starts at the same initial point x0 = x0,0 and runs toward the solution point
x∗ . In between we define further continuous Newton sequences, written as
{xi,k }, k = i, i + 1, . . ., which start at the discrete Newton iterates xij = xi,i
and also run toward x∗ . Note that the existence or even uniqueness of a
discrete solution point x∗j is not clear yet.
For the purpose of repeated induction, we assume that
xk−1
− x0 < ρ +
j
2
,
ω
which certainly holds for k = 1. In order to characterize the deviation between
discrete and continuous Newton sequences, we introduce the two majorants
ωΔxk ≤ hk ,
xkj − xk,0 ≤ k .
Recall from Theorem 2.2 that
hk+1 = 12 h2k .
(8.6)
For the derivation of a second majorant recursion, we apply the triangle
inequality in the form
xk+1,k+1 − xk+1,0 ≤ xk+1,k+1 − xk+1,k + xk+1,k − xk+1,0 .
The first term can be treated using assumption (8.5) so that
xk+1,k+1 − xk+1,k = xkj + Δxkj − xk,k + Δxk,k = Δxkj − Δxk,k ≤ δj .
For the second term, we may apply our nonlinear perturbation Lemma 8.1
(see the shaded regions in Fig. 8.1) to obtain
xk+1,k − xk+1,0 ≤ ω 12 xk,k − xk,0 + Δxk,0 xk,k − xk,0 .
Combining these results then leads to
xk+1,k+1 − xk+1,0 ≤ δj +
ω 2
+ hk k .
2 k
The above right side may be defined to be k+1 . Hence, together with (8.6),
we arrive at the following set of majorant equations
hk+1 =
1 2
h ,
2 k
k+1 = δj +
ω 2
+ hk k .
2 k
If we introduce the quantities αk = ωk + hk and δ = ωδj , we may obtain
the decoupled recursion
αk+1 = δ + 12 α2k ,
(8.7)
which can be started with α0 = h0 , since 0 = 0. Upon solving the equation
8.1 Asymptotic Mesh Independence
375
δ = α̂ − 12 α̂2 ,
we get the two equilibrium points
√
α̂1 = 1 − 1 − 2δ < 1 ,
α̂2 = 1 +
√
1 − 2δ > 1 .
Insertion into the recursion (8.7) then leads to the form
αk+1 − α̂ = 12 (αk − α̂)(αk + α̂) .
For αk < α̂2 we see that
1
(αk
2
+ α̂1 ) < 12 (α̂2 + α̂1 ) = 1 ,
which implies that
|αk+1 − α̂1 | < |αk − α̂1 | .
(8.8)
Hence, the fixed point α̂1 is attractive, whereas α̂2 is repelling. Moreover,
since αk + α̂1 > 0, we immediately obtain the result
sign(αk+1 − α̂) = sign(αk − α̂) .
Therefore, we have the following cases:
I.
II.
α0 ≤ α̂1 =⇒ αk ≤ α̂1 ,
α̂1 < α0 < α̂2 =⇒ α̂1 ≤ αk < α̂2 .
Insertion of the expressions for the used quantities then shows that cases I,II
directly correspond to cases I,II of the theorem. Its last asymptotic result is
an immediate consequence of (8.8). Finally, with application of the triangle
inequality
2
xk+1
− x0 ≤ k+1 + xk+1 − x0 < + ρ
j
ω
the induction and therefore the whole proof is completed.
We are, of course, interested whether a discrete solution point x∗j exists. The
above tracking theorem, however, only supplies the following result.
Corollary 8.3 Under the assumptions of Theorem 8.2, there exists at least
one accumulation point
∗
∗ 1
x̂j ∈ S (x , 2δj ) ∩ Xj ⊂ S x ,
∩ Xj ,
ω
which need not be a solution point of the discrete equations Fj (xj ) = 0.
376
8 PDE Boundary Value Problems
Proof. This is just the last asymptotic result of Theorem 8.2.
In order to prove more, Theorem 2.2 directs us to study the question of
whether an affine covariant Lipschitz condition holds for the finite dimensional mapping Fj , too.
Lemma 8.4 Let Theorem 2.2 hold for the mapping F : X → Y . For collinear
xj , yj , yj + vj ∈ Xj , define quantities wj ∈ Xj and w ∈ X according to
Fj (xj )wj = Fj (yj + vj ) − Fj (yj ) vj , F (xj )w = (F (yj + vj ) − F (yj )) vj .
Assume that the discretization method satisfies
w − wj ≤ σj vj 2 .
(8.9)
ωj ≤ ω + σ j
(8.10)
Then there exist constants
such that the affine invariant Lipschitz condition
wj ≤ ωj vj 2
holds for the discrete Newton process .
Proof. The proof is a simple application of the triangle inequality
wj ≤ w + wj − w ≤ ωvj 2 + σj vj 2 = (ω + σj ) vj 2 .
Finally, the existence of a unique solution x∗j is now an immediate consequence.
Corollary 8.5 Under the assumptions of Theorem 8.2 and Lemma 8.4 the
discrete Newton sequence {xkj }, k = 0, 1, . . . converges quadratically to a
unique discrete solution point
1
x∗j ∈ S (x∗ , 2δj ) ∩ Xj ⊂ S x∗ ,
∩ Xj .
ω
Proof. We just need to apply Theorem 2.2 to the finite dimensional mapping
Fj with the starting value x0j = x0 and the affine invariant Lipschitz constant
ωj from (8.10).
Remark 8.1
In the earlier papers [3, 4] two assumptions of the kind
Fj (xj )−1 ≤ βj ,
Fj (xj + vj ) − Fj (xj ) ≤ γj vj 8.1 Asymptotic Mesh Independence
377
have been made in combination with the uniformity requirements
βj ≤ β ,
γj ≤ γ .
(8.11)
Obviously, these assumptions lack any affine invariance. More important,
however, and as a consequence of the noninvariance, these conditions are
phrased in terms of operator norms, which, in turn, depend on the relation
of norms in the domain and the image space of the mappings Fj and F ,
respectively. For typical PDEs and norms we would obtain
lim βj → ∞ ,
j→∞
which clearly contradicts the uniformity assumption (8.11). Consequently, an
analysis in terms of βj and γj would not be applicable to this important case.
The situation is different with the affine covariant Lipschitz constants ωj :
they only depend on the choice of norms in the domain space. It is easy to
verify that
ωj ≤ βj γj .
From the above Lemma 8.4 we see that the ωj remain bounded in the limit
j −→ ∞, as long as ω is bounded—even if βj or γj blow up. Moreover, even
when the product βj γj remains bounded, the Lipschitz constant ωj may be
considerably lower, i.e.
ωj βj γj .
For illustration, just compare the simple R2 -example in Exercise 2.3.
Summarizing, we come to the following conclusion, at least in terms of upper
bounds: If the asymptotic properties
lim δj = 0 ,
j→∞
lim σj = 0
j→∞
can be shown to hold, then the convergence speed of the discrete ordinary
Newton method is asymptotically just the one for the continuous ordinary
Newton method—compare Exercises 8.3 and 8.4. Moreover, if related initial
guesses x0 and x0j and a common termination criterion are chosen, then even
the number of iterations will be nearly the same.
Bibliographical Remark. The ‘mesh independence’ principle has been
reported and even exploited for mesh design in papers by E.L. Allgower
and K. Böhmer [3] and S.F. McCormick [148]. Further theoretical investigations of the phenomenon have been given in the paper [4] by E.L. Allgower,
K. Böhmer, F.A. Potra, and W.C. Rheinboldt; that paper, however, lacked
certain important features, which have been discussed in Remark 8.1 above.
A first affine covariant theoretical study has later been worked out by
P. Deuflhard and F.A. Potra in [82]; from that analysis, the modified term
378
8 PDE Boundary Value Problems
‘asymptotic mesh independence’ naturally emerged. The presentation here
follows the much simpler and more intuitive treatment [198] of the topic as
given recently by M. Weiser, A. Schiela, and the author.
8.2 Global Discrete Newton Methods
In the present section we regard BVPs for nonlinear PDEs as given in already
discretized form on a fixed mesh, to be briefly called discrete PDEs here. In
what follows, we report about the comparative performance of exact and
inexact Newton methods in solving such problems. Part of the results are
from a recent paper by P. Deuflhard, U. Nowak, and M. Weiser [80].
In the exact Newton methods, we use either band mode LU -decomposition or
a sparse solver provided by MATLAB. Failure exits in the various numerical
tests are characterized by
• outmax: the Newton iteration (outer iteration in inexact Newton methods) does not converge within 75 iterations,
• itmax: the inner iteration per inexact Newton step does not converge
within itmax iterations,
• λ-fail: the adaptive damping strategy suggests some ‘too small’ damping
factor λk < 10−4 .
8.2.1 General PDEs
This section documents the comparative performance of residual based (or
affine contravariant) Newton methods versus error oriented (or affine covariant) Newton methods, both for the exact and the inexact versions, at a
common set of discrete PDE test problems.
Common test set. We consider a subset of the discrete PDE problems
given in [160]. In order to be able to compare exact and inexact methods, we
selected examples in only two space dimensions. This choice leads to moderate
system dimensions n that still permit a direct solution of the arising linear
equations. Throughout the examples, we use the usual second order, centered
finite differences on tensor product grids. Neumann boundary conditions are
included by simple one-sided differences, as usual.
Example 8.1 Artificial test problem (atp1). This problem comprises the
simple scalar PDE
Δu − (0.9 exp(−q) + 0.1u)(4x2 + 4y 2 − 4) − g = 0 ,
where
g = exp(u) − exp(exp(−q)) and q = x2 + y 2 ,
8.2 Global Discrete Newton Methods
379
with boundary conditions u|∂Ω = 0 on the domain Ω = [−3, 3]2 . The analytical solution is known to be u(x, y) = exp(−q).
Example 8.2 Driven cavity problems (dcp1000, dcp5000). This problem
involves the steady stream-function/vorticity equations
Δω + Re(ψx ωy − ψy ωx ) = 0 , Δψ + ω = 0 ,
where ψ is the stream-function and ω the vorticity. For the domain Ω = [0, 1]2
the following discrete boundary conditions are imposed (with Δx, Δy the
mesh sizes in x, y-direction)
∂ψ
∂y (x, 1)
= −16x2 (1 − x)2 ,
2
ω(x, 0) = − Δy
2 ψ(x, Δy) ,
∂ψ
2
ω(x, 1) = − Δy
2 [ψ(x, 1 − Δy) + Δy ∂y (x, 1)] ,
2
ω(0, y) = − Δx
2 ψ(Δx, y) ,
2
ω(1, y) = − Δx
2 ψ(1 − Δx, y) .
Problems dcp1000, dcp5000 correspond to Reynolds numbers Re = 1000, 5000,
respectively. For both cases the default initial guess is ψ 0 = ω 0 = 0.
As will be seen below, the residual based Newton strategy was unable to
solve problems dcp1000 and dcp5000 with this initial guess. That is why
we added problems dcp1000a and dcp5000a with the better initial guesses
ω 0 = y 2 sin(πx), ψ 0 = 0.1 sin(πx) sin(πy).
Example 8.3 Supersonic transport problem (sst2). The four model equations for the chemical species O, O3 , N O, N O2 , represented by the unknown
functions (u1 , u2 , u3 , u4 ), are
0 =
DΔu1 + k1,1 − k1,2 u1 + k1,3 u2 + k1,4 u4 − k1,5 u1 u2 − k1,6 u1 u4 ,
0 =
DΔu2 + k2,1 u1 − k2,2 u2 + k2,3 u1 u2 − k2,4 u2 u3 ,
0 =
DΔu3 − k3,1 u3 + k3,2 u4 + k3,3 u1 u4 − k3,4 u2 u3 + 800.0 + SST ,
0 =
DΔu4 − k4,1 u4 + k4,2 u2 u3 − k4,3 u1 u4 + 800.0 ,
where D = 0.5 · 10−9 ,
k1,1 , . . . , k1,6 = 4 · 105 , 272.443800016, 10−4, 0.007, 3.67 · 10−16 , 4.13 · 10−12 ,
k2,1 , . . . , k2,4 = 272.4438, 1.00016 · 10−4 , 3.67 · 10−16 , 3.57 · 10−15 ,
k3,1 , . . . , k3,4 = 1.6 · 10−8 , 0.007, 4.1283 · 10−12, 3.57 · 10−15,
k4,1 , . . . , k4,3 = 7.000016 · 10−3 , 3.57 · 10−15 , 4.1283 · 10−12 , and
3250 if (x, y) ∈ [0.5, 0.6]2
SST =
360
otherwise.
380
8 PDE Boundary Value Problems
Homogeneous Neumann boundary conditions are imposed on the unit square.
For the initial guess we take
u01 (x, y) = 109 , u02 (x, y) = 109 , u03 (x, y) = 1013 , u04 (x, y) = 107 .
Again we consider better initial guesses to allow for convergence in the residual based Newton methods:
u0i → (1 + 100(sin(πx) sin(πy))2 )u0i .
Name
atp1
dcp1000
dcp1000a
dcp5000
dcp5000a
sst2
sst2a
Grid
31 × 31
31 × 31
31 × 31
63 × 63
63 × 63
51 × 51
51 × 51
Dim n
961
1922
1922
7983
7983
10404
10404
OrdNew
4
outmax
9
outmax
outmax
outmax
outmax
Table 8.1. Test set characteristics.
Characteristics of the selected test set are arranged in Table 8.1. In order
to give some idea about the complexity of the individual problems, we first
applied an exact ordinary Newton method (uncontrolled)—see the last column of the table. All of its failures are due to ‘too many’ Newton (outer)
iterations (recall outmax= 75).
Exact Newton methods. Recall that exact Newton methods require the
direct solution of the arising linear subsystems for the Newton corrections.
Hence, adaptivity only shows up through affine invariant trust region (or
damping) strategies. From the code family NLEQ we compare the following
variants:
• NLEQ-RES requiring monotonicity in the residual norm F 2 , as discussed
in Section 3.2.2,
• NLEQ-RES/L requiring monotonicity in the preconditioned residual norm
CL F 2 , also discussed in Section 3.2.2; the preconditioner CL comes from
incomplete LU -decomposition with fill-in only accepted within the block
pentadiagonal structure (compare, e.g., [184]), and
• NLEQ-ERR requiring monotonicity in the natural level function, as discussed
in Section 3.3.3.
8.2 Global Discrete Newton Methods
381
The residual based methods realize the restricted monotonicity test (3.32).
For termination, the (possibly preconditioned) criterion (2.70) with FTOL =
10−8 has been taken, except for the badly scaled problems sst, which required
FTOL = 10−5 to terminate within a tolerable computing time. The error oriented methods realize the restricted monotonicity test (3.47) and the (scaled)
termination criterion (2.14) with XTOL = 10−8 .
Name
atp1
dcp1000
dcp1000a
dcp5000
dcp5000a
sst2
sst2a
NLEQ-RES
4 (0)
outmax
21 (17)
outmax
42 (39)
λ-fail
38 (33)
NLEQ-RES/L
4 (0)
10 (5)
8 (2)
outmax
λ-fail
12 (11)
15 (13)
NLEQ-ERR
4 (0)
8 (4)
8 (2)
11 (7)
8 (2)
13 (8)
19 (14)
Table 8.2. Exact Newton codes: adaptive control via residual norm (NLEQ-RES),
preconditioned residual norm (NLEQ-RES/L), and error norm (NLEQ-ERR).
In Table 8.2 we compare the residual based versus the error oriented exact
Newton codes in terms of Newton steps (in parentheses: damped). As can
be seen, there is striking evidence that the error oriented adaptive Newton
methods are clearly preferable to the residual based ones, at least for the
problem class tested here.
The main reason for this phenomenon is certainly that the arising discrete
Jacobian matrices are bound to be ill-conditioned, the more significant the
finer the mesh is. For this situation, the limitation of residual monotonicity
has been described at the end of Section 3.3.1. Example 3.1 has given an
illustration representative for a class of ODE boundary value problem. The
experimental evidence here seems to indicate that the limitation carries over
to PDE boundary value problems as well.
Inexact Newton methods. Finite dimensional inexact Newton methods
contain some inner iterative solver, which induces the necessity of an accuracy matching between inner and outer iteration. The implemented ILUpreconditioning [184] is the same as in the exact Newton codes above. In the
code family GIANT, various affine invariant damping and accuracy matching
strategies are realized—according to the selected affine invariance class. The
failure exit itmax was activated at 2000 inner iterations.
Residual based methods. For this type of inexact Newton method, we chose
the codes GIANT-GMRES/R and GIANT-GMRES/L with right (R) or left (L) preconditioning.
As a first test, we selected the standard convergence mode from Sections 2.2.4
and 3.2.3, prescribing ηk ≤ η̄ with threshold values η̄ = 0.1 and η̄ = 0.001.
382
8 PDE Boundary Value Problems
Name
atp1
dcp1000
dcp1000a
dcp5000
dcp5000a
sst2
sst2a
NLEQ-RES
4 (0)
outmax
21 (17)
outmax
42 (39)
λ-fail
38 (33)
GIANT-GMRES/R/0.001
4 (0)
34
outmax
21 (17)
788
outmax
43 (39)
5021
15 (10)
1376
itmax
GIANT-GMRES/R/0.1
4 (1)
28
outmax
28 (23)
605
outmax
58 (53)
3208
itmax
itmax
Table 8.3. Residual based Newton codes: exact version NLEQ-RES versus inexact
version GIANT-GMRES/R for threshold values η̄ = 0.001 and η̄ = 0.1.
Name
atp1
dcp1000
dcp1000a
dcp5000
dcp5000a
sst2
sst2a
NLEQ-RES/L
4 (0)
10 (5)
8 (2)
outmax
λ-fail
12 (10)
15 (13)
GIANT-GMRES/L/0.001
4 (0)
31
10 (5)
380
8 (1)
279
itmax
24 (15)
1700
15 (12)
252
18 (15)
465
GIANT-GMRES/L/0.1
4 (1)
25
16 (10)
309
12 (3)
229
outmax
outmax
outmax
outmax
Table 8.4. Preconditioned residual based Newton codes: exact version NLEQ-RES/L
versus inexact version GIANT-GMRES/L for threshold values η̄ = 0.001 and η̄ = 0.1.
In Table 8.3, we compare exact versus inexact Newton methods, again at the
common test set, in terms of Newton steps (in parentheses: damped Newton
steps) and inner iterations. For comparison, the first column is identical to the
first one from Table 8.2. In Table 8.4, the performance of two GIANT-GMRES/L
versions is documented. This time, the first column is the second one from
Table 8.2.
As can be seen from both tables, the inexact Newton codes behave very
much like their exact counterparts in terms of outer iterations, with erratic
discrepancies now and then. In view of the anyway poor behavior of the
residual based Newton methods in this problem class, we did not realize the
fully adaptive accuracy matching strategy (linear or quadratic convergence
mode) in the frame of residual based inexact Newton methods.
Error oriented Newton methods. For this type of inexact Newton method, we
chose the codes GIANT-CGNE/L and GIANT-GBIT/L, both with left (L) preconditioning. Adaptive matching strategies as worked out in Sections 2.1.5
and 3.3.4 have been realized. Initial values for the arising inner iterations
were chosen according to the nested suggestions (3.59) and (3.60). Note that
these inexact codes realize a damping strategy and a termination criterion
slightly different from those in NLEQ-ERR. In view of a strict comparison, we
8.2 Global Discrete Newton Methods
383
constructed an exact variant NLEQ-ERR/I, which realizes just these modifications, i.e. which is an inexact Newton-ERR code with exact inner solution.
Name
atp1
dcp1000
dcp1000a
dcp5000
dcp5000a
sst2
sst2a
NLEQ-ERR/I
5 (0)
10 (5)
9 (3)
13 (8)
10 (3)
15 (11)
20 (15)
GIANT-CGNE/L
5 (0)
237
itmax
itmax
itmax
itmax
16 (11)
23084
20 (15)
39889
GIANT-GBIT/L
5 (0)
122
10 (5) 1388
9 (3) 2241
14 (8) 5943
10 (3) 9504
16 (11) 1549
20 (15) 2399
Table 8.5. Error oriented Newton codes: exact version NLEQ-ERR/I versus inexact
versions GIANT-CGNE/L and GIANT-GBIT/L for threshold values δ̄ = 10−3 .
In Table 8.5, we give results for the ‘standard convergence mode’, imposing
the (scaled) condition δk ≤ δ̄ for the threshold value δ̄ = 10−3 throughout,
as defined in (2.61) for the local Newton method and (3.55) for the global
Newton method, the latter via ρ = 2δ̄/(1 − 2δ̄). As can be seen, the first
column for NLEQ-ERR/I and the third column of Table 8.2 for NLEQ-ERR
differ only marginally.
From these numerical experiments, we may keep the following information:
• The error estimator (1.31) for CGNE is more reliable than (1.37) for GBIT.
• Nevertheless CGNE is less efficient than GBIT—compare Remark 1.2.
• The code GIANT-GBIT/L essentially reproduces the outer iteration pattern
of the exact Newton code NLEQ-ERR∗.
More insight into GIANT-GBIT/L can be gained from Table 8.6 where we compare the ‘standard convergence mode’ (SM) with the ‘quadratic convergence
mode’ (QM), again in terms of outer (damped) and inner iterations. Different
sets of control parameters are applied. The parameter ρ defines δ̄ = 12 ρ/(1+ρ)
via (3.55). As a default, the parameter ρ is fixed to ρ = 12 ρ, which, in turn,
defines ρmax = ρ/(1 + ρ) via (3.70).
The first column, SM(.025, .05), presents results obtained over our common
test set, when the accuracy matching strategies (3.66) with (3.71) and (3.50)
with (3.55) are implemented; the values (
ρ, ρ) = (.025, .05) represent the
largest values, for which all problems from the common test set were still
solvable. This column should be compared with the third column in Table 8.5,
where GIANT-GBIT/L has been realized roughly in an SM(.001, .002) variant:
considerable savings are visible.
Detailed examination of the numerical results has revealed that the weakest
point of this algorithm is the rather poor error estimator (1.37) realized
384
8 PDE Boundary Value Problems
Name
atp1
dcp1000
dcp1000a
dcp5000
dcp5000a
sst2
sst2a
SM(.025, .05)
5 (0)
91
11 (5)
904
11 (3) 1272
19 (11) 3952
16 (3) 4304
22 (13)
963
25 (16) 1429
SM∗ (1/16, 1/8)
5 (0)
66
12 (5)
817
12 (4)
1458
16 (8)
3417
11 (1)
3475
19 (12)
1037
24 (17)
1597
QM(.025, .05)
5 (0)
97
10 (5)
852
9 (3) 1180
16 (11) 3802
10 (3) 3539
18 (13)
842
22 (16) 1365
Table 8.6. Comparison of different variants of error oriented inexact Newton code
GIANT-GBIT/L. Accuracy matching strategies SM(e
ρ, ρ) and QM(e
ρ, ρ) for control
parameters (e
ρ, ρ). SM∗ realizes an exact computation of the inner iteration error.
within GBIT, which is often quite off scale. For an illustration of this effect,
the second column presents results for version SM∗ , which realizes a precise
error estimator
i = δxki − Δxk instead of the unsatisfactory estimator (1.37); in this case, the relaxed choice
(
ρ, ρ) = (1/16, 1/8) appeared to be possible. Consequently, savings of inner
iterations can be observed.
These savings, however, are not too dramatic when compared with the version
QM shown in the third column; here the quadratic accuracy matching rule
(3.56) is realized, which does not differ too much from the rule (2.62). For
the control parameters we again selected (
ρ, ρ) = (.025, .05) to allow for a
comparison with SM in the first column. Obviously, this column shows the
best comparative numbers.
Summary. From our restricted set of numerical experiments, we may nevertheless draw certain practical conclusions:
• Among the inner iterations for an inexact Newton-ERR method, the algorithm GBIT is the clear ‘winner’—despite the rather poor computational
estimator for the inner iteration error, which is presently realized.
• Linear preconditioning also plays a role in nonlinear preconditioning as
realized in the inexact Newton-ERR codes; in particular, the better the
linear preconditioner, the better the inner iteration error estimator, the
better the performance of the whole inexact Newton-ERR method.
• The ‘quadratic convergence mode’ in the local convergence phase can save
a considerable amount of computing time over the ‘standard convergence
mode’.
8.2 Global Discrete Newton Methods
385
8.2.2 Elliptic PDEs
This section documents the comparative performance of exact versus inexact
affine conjugate Newton methods at a common set of nonlinear BVPs for elliptic discrete PDEs. Recall that elliptic PDEs are associated with underlying
convex optimization problems—see Sections 2.3 and 3.4.
Test set. Below three discretized nonlinear elliptic PDE BVPs in two space
dimensions are given. Of course, their corresponding discretized functional
is also at hand. All discretizations are simple finite difference schemes on
uniform meshes.
Example 8.4 Simple elastomechanics problem (elas). For Ω =]0, 1[2 , minimize the functional (total energy in Ogden material law)
F 2 + (detF )−1 − M (1/2, −1)u dx with F = I + ∇u .
Ω
Homogeneous Dirichlet conditions on the boundary part {0}×[0, 1] and natural boundary conditions on the remaining boundary part are imposed. Physically speaking, u(x) ∈ R2 is the displacement of an elastic body. The volume
force (1/2, −1)T acting on the body is scaled by M , which can be used to
weight the ‘nonlinearity’ of the problem. As initial value we chose u0 = 0 in
agreement with the Dirichlet conditions.
Detailed examination reveals that the above functional is not globally convex
on the whole domain of definition, but only in a neighborhood of the solution.
Fortunately, for the given initial guesses, our Newton codes did not encounter
any nonpositive second derivatives. The locally unique solution is depicted in
Figure 8.2, right.
Example 8.5 Minimal surface problem over convex domain (msc). Given
Ω =]0, 1[2 , minimize the surface area
1
(1 + |∇u|2 ) 2 dx
Ω
subject to the Dirichlet boundary conditions
u(x1 , x2 ) = M (x1 + (1 − 2x1 )x2 ) on ∂Ω .
Here u(x) ∈ R is the vertical position of the surface parametrized over Ω.
The scaling parameter M of the boundary conditions allows to weight the
‘nonlinearity’ of the problem. The initial value u0 is chosen as the bilinear
interpolation of the boundary conditions. This problem has a unique welldefined solution depicted in Figure 8.2, left.
386
8 PDE Boundary Value Problems
1
0.8
10
0.6
8
0.4
6
0.2
4
0
2
ï0.2
0
1
ï0.4
0.8
ï0.6
0.6
1
0.8
0.4
ï0.8
0.6
0.4
0.2
0
0.2
ï1
ï0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Fig. 8.2. Left: solution of problem msc (M = 10, h = 1/63). Right: solution of
problem elas (M = 2, h = 1/31).
Example 8.6 Minimal surface problem over nonconvex domain (msnc).
Given the domain Ω =]0, 2[2 \]1, 2[2 , minimize the surface area
1
(1 + |∇u|2 ) 2 dx
Ω
subject to the Dirichlet boundary conditions
u = 0 on [0, 2] × {0} ∪ {0} × [0, 2] ,
u = M on [1, 2] × {1} ∪ {1} × [1, 2] .
On the remaining boundary parts, [0, 1] × {2} ∪ {2} × [0, 1], homogeneous
Neumann boundary conditions ∂n u = 0 are imposed. Here u(x) ∈ R is the
vertical position of the surface parameterized over Ω. The scaling parameter
M plays the same role as in problem msc. The initial value u0 is chosen to
be the linear interpolation of the Dirichlet boundary conditions on [0, 1] ×
[1, 2]∪[1, 2]×[0, 1] and the bilinear interpolation of the thus defined boundary
values on [0, 1]2 .
This problem has been deliberately constructed such that the underlying
PDE does not have a unique continuous solution. Indeed, function space
Newton multilevel methods (to be presented in Section 8.3 below) are able
to detect this nonexistence: even though there exists a finite dimensional
‘pseudosolution’ on each mesh with size h, the local convergence domain of
Newton’s method shrinks when h → 0. In the present setting of discrete
PDEs, however, Newton’s method will just supply a discrete solution on
each of the meshes. As shown in Figure 8.3, these discrete solutions exhibit
an interior ‘discrete discontinuity’, which is the sharper, the finer the mesh
is.
8.2 Global Discrete Newton Methods
387
2
1.8
2
1.6
1.8
1.4
1.6
1.2
1.4
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
2
2
0.2
1.8
0.2
1.8
1.6
0
1.4
0
0.2
1.2
0.4
0.8
1.4
0
1.2
0.2
0.4
1
0.6
1.6
0
1
1.4
0.8
0.8
1
0.6
1.2
1
0.6
0.8
0.6
1.2
0.4
1.6
2
0.4
1.4
1.6
0.2
1.8
0.2
1.8
0
2
0
Fig. 8.3. Discrete solutions of problem msnc. Left: M = 2, h = 1/8, right:
M = 2, h = 1/32.
Table 8.7 gives some ‘measure’ of the problem complexity for selected problem
sizes of low dimension n: the value Mmax in the last column indicates the
maximal nonlinearity weight factor, for which the ordinary Newton method
(uncontrolled) had still converged in our tests.
Name
msc
elas
msnc
Grid
32 × 32
32 × 32
32 × 32
Dim n
1024
2048
3072
Mmax
6.2
1.0
1.9
Table 8.7. Test set characteristics for special 2D grid.
Incidentally, below we also treat much larger problems, where dimensions up
to n ≈ 200.000 arise.
Exact versus inexact Newton methods. For the exact as well as the
inexact Newton iteration, the energy error termination criterion (2.110) with
ETOL = 10−8 is taken. For Newton-PCG methods, we use the inner iteration
termination criterion (1.25) and the accuracy matching strategy as worked
out in Sections 2.3.3 and 3.4.3. As preconditioners we tested both the Jacobi
and the incomplete Cholesky preconditioner (ICC) provided by MATLAB
(with droptol = 10−3 ). The failure exit itmax was activated at more than
500 inner iterations.
Local versus global Newton methods. In Table 8.8, we give comparative results for varying weight factor M at problem msc. Among the local Newton methods, we deliberately included the rather popular simplified Newton
388
8 PDE Boundary Value Problems
method (with initial Jacobian throughout the iteration)—see Section 2.3.2.
The Newton-PCG algorithms are run in the quadratic convergence mode—see
Section 2.3.3. The following features can be clearly observed:
• The local Newton methods converge only for the mildly nonlinear case
(here: small M ).
• Among the local Newton methods, the simplified variant behaves poorest.
• Exact and inexact Newton methods realize nearly the same number of
(outer) Newton iterations, both local and global.
M
2
5
10
simplified
21
DIV
DIV
local
exact inexact
5
5
7
7
DIV
DIV
exact
9
10
10
global
inexact
9
9
10
Table 8.8. Problem msc: comparative Newton steps (DIV: divergence).
Asymptotic mesh independence. In Table 8.9, we test different discrete Newton algorithms over the whole test set for decreasing mesh sizes. Asymptotic
mesh independence as studied in Section 8.1 (see also Exercise 8.4) is clearly
visible in problems msc and elas, but not in problem msnc, which does not
have a unique continuous solution (see also Table 8.11 in Section 8.3.2 below). In the latter problem failures occur on the finest meshes—in agreement
with the subsequent Example 8.9. The missing entries indicate the fact that
the inexact codes were able to tackle much larger problems than the exact
ones—both due to time and, even more pronounced, memory requirements
of the direct solver on the finer meshes.
N
4
8
16
32
64
128
256
msc(M = 10)
exact inexact
9
8
10
9
10
9
10
10
10
10
10
10
elas(M = 2)
exact inexact
10
9
10
10
10
10
10
10
11
msnc(M = 2)
exact inexact
9
8
9
9
10
10
10
11
13
λ-fail
itmax
Table 8.9. Test set: Newton steps for decreasing mesh sizes h = 1/N .
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs
389
Different preconditioners. In Table 8.10, the Jacobi preconditioner (Jac) and
the incomplete Cholesky preconditioner (ICC) are compared for the quadratic
and the linear convergence mode. For the linear convergence mode, Θ̄ = 0.5
has been chosen. As can be seen, Jac is insufficient for fine meshes. ICC is
more effective, at least for small up to moderate size meshes. The linear convergence mode is comparable to the quadratic convergence mode—as opposed
to the behavior in the function space Newton method presented in Section
8.3 below.
msc
(M=3.5)
elas
(M=0.2)
n
4
8
16
32
64
128
256
512
4
8
16
32
64
128
7
6
6
6
6
6
6
6
6
5
5
5
5
6
quadratic
ICC
Jac
(16) 7 (39)
(15) 6 (134)
(19) 7 (385)
(25) 8 (921)
(35)
itmax
(57)
itmax
(103)
itmax
(210)
itmax
(18) 6 (174)
(19) 6 (479)
(29)
itmax
(44)
itmax
(80)
itmax
(176)
itmax
linear
ICC
7 (14)
6 (12)
6 (12)
7 (16)
9 (24)
12 (52)
15 (96)
19 (235)
6 (12)
6 (12)
7 (18)
9 (36)
11 (67)
14 (144)
Jac
8 (30)
15 (120)
Θ≥1
Θ≥1
Θ≥1
Θ≥1
Θ≥1
Θ≥1
Θ≥1
Θ≥1
Θ≥1
Θ≥1
Θ≥1
itmax
Table 8.10. Local inexact Newton-PCG method: comparative outer (inner)
iterations. Quadratic versus linear convergence mode, Jacobi (Jac) versus incomplete Cholesky (ICC) preconditioning.
Summary. For elliptic discrete nonlinear PDEs both the exact and the
inexact affine conjugate Newton methods perform efficiently and reliably, in
close connection with the associated convergence theory. The inexact Newton
code GIANT-PCG with ICC preconditioning seems to be a real competitor
to so-called nonlinear PCG methods (for references see Section 2.3.3).
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs
In this section we consider minimization problems of the kind
f (x) = min ,
390
8 PDE Boundary Value Problems
wherein f : D ⊂ X → R is assumed to be a strictly convex C 2 -functional
defined on an open convex subset D of a Banach space X. Let X be endowed with a norm · . In order to assure the existence of a minimum point
x∗ ∈ D, we assume that X is reflexive; in view of the subsequent finite element
method (FEM), we choose X = W 1,p for 1 < p < ∞. Moreover, for given initial guess x0 ∈ D, we assume that the level set L0 := {x ∈ D|f (x) ≤ f (x0 )}
is nonempty, closed, and bounded. Under these assumptions the existence of
a unique minimum point x∗ is guaranteed. In this case the nonlinear minimization problem is equivalent to the nonlinear operator equation
F (x) := f (x) = 0 , x ∈ D .
(8.12)
In the present section, this equation is understood to be a nonlinear elliptic
PDE problem. In order to guarantee the feasibility of Newton’s method, we
further assume that the PDE problem is strictly elliptic, which means that
its symmetric Frechét-derivative F (x) = f (x) is strictly positive.
In abstract notation, the ordinary Newton method for the mapping F reads
(k = 0, 1, . . .)
F (xk )Δxk = −F (xk ) , xk+1 = xk + Δxk ,
which just describes the successive linearization, often also called quasilinearization, of the above nonlinear operator equation. Since equation (8.12) is a
nonlinear elliptic PDE in the Banach space W 1,p , the above Newton sequence
consists of solutions of linear elliptic PDEs in some Hilbert space, say Hk ,
associated with each iterate xk ∈ W 1,p . For reasonable arguments x, there
exist energy products ·, F (x)·, which induce energy norms ·, F (x)·1/2 .
The question of whether these energy norms are bounded for all arguments
of interest needs to be discussed inside the proofs of the theorems to be stated
below.
In the subsequent analysis, we will ‘lift’ these energy products and energy
norms from H to W 1,p (in the sense of dual pairing) defining the corresponding local energy products ·, F (x)· as symmetric bilinear forms and
the induced local energy norms ·, F (x)·1/2 for arguments x in appropriate
subsets of W 1,p . Moreover, motivated by the notation in Hilbert space, where
the operator F (x)1/2 is readily defined, we also adopt the shorthand notation
F (x)1/2 · ≡ ·, F (x)·1/2
to be only used in connection with the local energy norms.
As already mentioned for space-like ODE BVPs in Section 7.4.2, quasilinearization, here for BVPs in more than one space dimension, cannot be realized without approximation errors. This means that we need to study inexact
Newton methods
F (xk ) δxk = −F (xk ) + rk ,
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs
391
equivalently written as
F (xk ) δxk − Δxk = rk .
Here the discretization errors show up either as residuals rk or as the discrepancy between the inexact Newton corrections δxk and the exact Newton
corrections Δxk . Among the discretization methods, we will focus on Galerkin
methods, known to satisfy the Galerkin condition
δxk , F (xk )(δxk − Δxk ) = δxk , rk = 0 .
(8.13)
Such a condition also holds in the finite dimensional inexact Newton-PCG
method, where the residuals originate from the use of PCG as inner iterative
solver—just look up condition (2.98) in Section 2.3.3. Recall that Newton-PCG
methods are relevant for the discrete PDE situation, as presented in Section
8.2.2. Here, however, we want to treat the infinite dimensional exact Newton method approximated by an adaptive finite dimensional inexact Newton
method. The benefit to be gained from adaptivity will become apparent in
the following.
8.3.1 Local Newton-Galerkin methods
In this section we study the ordinary Newton-Galerkin method
xk+1 = xk + δxk ,
where the iterates xk are in W 1,p and the inexact Newton corrections δxk
satisfy (8.13). With the above theoretical considerations we are ready to just
modify the local convergence theorem for Newton-PCG methods (Theorem
2.20) in such a way that it covers the present infinite dimensional setting.
Theorem 8.6 Let f : D → R be a strictly convex C 2 -functional to be minimized over some open convex domain D ⊂ W 1,p endowed with the norm · .
Let F (x) = f (x) be strictly positive. For collinear x, y, z ∈ D, assume the
affine conjugate Lipschitz condition
−1/2 F (z)
F (y) − F (x) v ≤ ω F (x)1/2 (y − x) · F (x)1/2 v for some 0 ≤ ω < ∞. Consider an ordinary Newton-Galerkin method satisfying (8.13) with approximation errors bounded by
δk :=
F (xk )1/2 (δxk − Δxk )
.
F (xk )1/2 δxk At any well-defined iterate xk , define the exact and inexact energy error
norms by
392
8 PDE Boundary Value Problems
k = F (xk )1/2 Δxk 2 ,
δk = F (xk )1/2 δxk 2 =
k
1 + δk2
and the associated Kantorovich quantities as
hk = ω F (xk )1/2 Δxk ,
hδk = ω F (xk )1/2 δxk =
hk
.
1 + δk2
For given initial guess x0 ∈ D assume that the level set
L0 := {x ∈ D | f (x) ≤ f (x0 )}
is nonempty, closed, and bounded. Then the following results hold:
I. Linear convergence mode. Assume that x0 satisfies
h0 ≤ 2Θ < 2
(8.14)
for some Θ < 1. Let δk+1 ≥ δk throughout the inexact Newton iteration.
Moreover, let the Galerkin approximation be controlled such that
δ
δ
δ 2
hk + δk hk + 4 + (hk )
ϑ(hδk , δk ) =
≤ Θ.
(8.15)
2 1 + δk2
Then the iterates xk remain in L0 and converge at least linearly to the minimum point x∗ ∈ L0 such that
and
F (xk+1 )1/2 Δxk+1 ≤ Θ F (xk )1/2 Δxk (8.16)
F (xk+1 )1/2 δxk+1 ≤ Θ F (xk )1/2 δxk .
(8.17)
II. Quadratic convergence mode. Let, for some ρ > 0, the initial guess
x0 satisfy
2
h0 <
(8.18)
1+ρ
and the Galerkin approximation be controlled such that
δk ≤
ρhδ
k
.
hδk + 4 + (hδk )2
(8.19)
Then the inexact Newton iterates xk remain in L0 and converge quadratically
to the minimum point x∗ ∈ L0 such that
ω
F (xk+1 )1/2 Δxk+1 ≤ (1 + ρ) F (xk )1/2 Δxk 2
(8.20)
2
and
ω
F (xk+1 )1/2 δxk+1 ≤ (1 + ρ) F (xk )1/2 δxk 2 .
(8.21)
2
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs
393
III. Functional descent. The convergence in terms of the functional can
be estimated by
− 61 hδk δk ≤ f (xk ) − f (xk+1 ) − 12 δk ≤ 16 hδk δk .
The proof in the general Newton-Galerkin case is—mutatis mutandis—the
same as the one for the more special Newton-PCG case in Theorem 2.20. In
passing we mention that the above discussed boundedness of the local energy norms and, via the Cauchy-Schwarz inequality, also of the local energy
products is actually guaranteed by (8.14), (8.16), and (8.17) in the linear convergence mode or by (8.18), (8.20), and (8.21) in the quadratic convergence
mode.
For linear elliptic PDEs, we have computational approximation error estimates available, typically incorporated within adaptive multilevel FEM (Section 1.4.5), which are a special case of Galerkin methods. Hence, we may
readily satisfy the above threshold criteria (8.15) or (8.19), respectively. Thus
we are only left with the decision of whether to use the linear or the quadratic
convergence mode in such a setting—an important algorithmic question that
deserves special attention.
Computational complexity model. In order to get some insight, we study
a rather simple, but nevertheless meaningful complexity model. It starts from
the fact that at the final iterate, say xq , we want to meet the prescribed energy
error tolerance criterion (2.110), i.e.,
.
q = ETOL2 .
If we replace the absolute error parameter ETOL2 0 by a relative error
parameter EREL 1 with ETOL2 = EREL2 ·0 , then we may rewrite the
above final accuracy requirement as
.
Θ0 · Θ1 · · · Θq−1 = EREL ,
which is equivalent to
q−1
log
k=0
1 .
1
= log
.
Θk
EREL
(8.22)
The number q of Newton steps is unknown in advance. Let Ak denote the
amount of work for step k. Then we will want to minimize the total amount
of work, i.e.,
q
Atotal =
Ak = min
k=0
subject to the constraint (8.22). For the solution of this discrete optimization problem, there exists a quite efficient established heuristics, the popular
394
8 PDE Boundary Value Problems
greedy algorithm—see, e.g., Chapter 9.3 in the introductory textbook [2] by
M. Aigner. From this, we obtain the prescription that, at Newton step k, the
algorithm should maximize the information gain per unit work, i.e.,
Ik =
1
1
log
= max .
Ak
Θk
(8.23)
In order to maximize this quantity with respect to the variable δk , the general
relation (2.109) is applicable, which reads
Θk ≤ ϑ(hδk , δk ) .
To simplify matters, we study the case hk → 0 here. Thus we arrive at the
rough model
δk
.
Θk = ϑ(0, δk ) =
,
1 + δk2
which, in view of (8.23), is equivalent to
1
1
log
∼ log 1 + 2 .
Θk
δk
Next we compare two variants of Newton-Galerkin methods, the finite dimensional case (PCG) and the infinite dimensional case (FEM), which differ
in the amount of work Ak as a function of δk .
Inexact Newton-PCG method for discrete PDEs. Assume that we attack a
nonlinear discrete elliptic PDE by some inexact Newton method with PCG as
inner iteration—as in the algorithm GIANT-PCG. This is exactly the situation
treated in Section 8.2.2. For system dimension n, we have to consider
• the evaluation of the Jacobian matrix J = F (xk ), which is typically sparse,
so that an amount O(n) needs to be counted,
• the work per PCG step (evaluation of inner products), which for the sparse
Jacobian J is also O(n),
• the number mk of PCG iterations at Newton step k: with preconditioner
B we have (compare, e.g., Corollary 8.18 in the textbook [77])
mk ∼
κ(BJ) log 2 1 + 1/δk2 .
Summing up, we arrive at the rough estimate
Ak ∼ c1 + c2 log 1 + 1/δk2 n ∼ const + log 1 + 1/δk2 ,
where ‘const’ represents some positive constant. So we finally end up with
Ik ∼
log(1 + 1/δk2 )
= max .
const + log(1 + 1/δk2 )
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs
395
The right hand side is a monotone decreasing function of δk , which directs us
towards the smallest possible value of δk , i.e., to the quadratic convergence
mode. It may be worth noting that the above analysis would lead to the same
decision, if PCG were replaced by some linear multigrid method.
Inexact Newton multilevel FEM for continuous PDEs. For the inner iteration
we now take an adaptive multilevel method for linear elliptic PDEs (such as
the multiplicative multigrid algorithm UG by G. Wittum, P. Bastian et al. [22]
or the additive multigrid algorithm KASKADE by P. Deuflhard, H. Yserentant
et al. [78, 36, 23]). An example of such an algorithm is implemented in our
code Newton-KASKADE. As a consequence of the adaptivity, the dimension
n of subproblems to be solved at step k depends on δk . Let d denote the
underlying spatial dimension. At iteration step k on refinement level j of
the multilevel discretization, let njk be the number of nodes and jk the local
energy. With l = lk we mean the final discretization level, at which the
prescribed final accuracy δk is achieved. On energy equilibrated meshes for
linear elliptic PDEs, we have the following asymptotic theoretical result (see
I. Babuška et al. [13])
n0k
nlk
2/d
∼
l
∞
δk2
k − k
≤
.
∞
1 + δk2
k
Any decent multigrid solver for linear elliptic PDEs will require an amount
of work proportional to the number of nodes, i.e.
d/2
Ak ∼ nlk ∼ n0k 1 + 1/δk2
.
Inserting this result into Ik , we arrive at the rough estimate
−d/2
Ik ∼ 1 + 1/δk2
log 1 + 1/δk2 = max .
For variable space dimension d this scalar function has its maximum at
δk = 1/ exp(2/d) − 1 ,
which, with the help of (2.113), then leads to the choice
Θ = exp(−1/d) .
We thus have the approximate values
d=2:
δk = 0.76, Θ = 0.61 ,
d=3:
δk = 1.03, Θ = 0.72 .
Even though our rough complexity model might not cover such large values of
δk , these results may nevertheless be taken as a clear indication to favor the
linear over the quadratic convergence mode in an adaptive multilevel setting.
Empirical tests actually suggested to use δk ≈ 1 corresponding to Θ ≈ 0.7 as
default values.
396
8 PDE Boundary Value Problems
Example 8.7 By modification of a problem given by R. Rannacher [175], we
consider the convex functional in two space dimensions (with x, y Euclidean
coordinates here)
f (u) =
1 + |∇u|2
p/2
− gu dx , p > 1 , x ∈ Ω ⊂ R2 , u ∈ W 1,p (Ω)
Ω
for the specification p = 1.4, g ≡ 0. The functional gives rise to the first and
second order expressions
F (u), v
p(1 + |∇u|2 )p/2−1 ∇u, ∇v − gv dx ,
=
Ω
w, F (u)v =
Ω
p (p − 2)(1 + |∇u|2 )p/2−2 ∇w, ∇u ∇u, ∇v
+(1 + |∇u|2 )p/2−1 ∇w, ∇v dx .
With ·, · the Euclidean inner product in R2 , the term v, F (u)v is strictly
positive for p ≥ 1.
Fig. 8.4. Example 8.7. Newton-KASKADE iterates: Top: initial guess u0
on initial coarse grid, bottom: iterate u3 on automatically refined grid. Thick lines:
homogeneous Dirichlet boundary conditions and level lines, thin lines: Neumann
boundary conditions.
In order to solve this problem, we used the linear convergence mode in an
adaptive Newton multilevel FEM with KASKADE to solve the arising linear
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs
397
elliptic PDEs. Figure 8.4 compares the starting guess u0 on its coarse mesh
(j = 1) with the inexact Newton iterate u3 on its fine mesh (j = 14). The
coarse mesh consists of n1 = 17 nodes, the fine mesh of n14 = 2054 nodes;
for comparison, a uniformly refined mesh at level j = 14 would have about
n14 ≈ 136000 nodes. Note that—in the setting of multigrid methods, which
require O(n) operations—the total amount of work would be essentially blown
up by the factor n14 /n14 ≈ 65. Apart from this clear computational saving,
adaptivity also nicely models the two critical points on the boundary, the
re-entrant corner and the discontinuity point.
8.3.2 Global Newton-Galerkin methods
In this section we study the inexact global Newton-Galerkin method
xk+1 = xk + λk δxk
in terms of iterates xk ∈ W 1,p , inexact Newton corrections δxk satisfying
(8.13), and damping factors λk to be chosen appropriately. As the most
prominent representatives of such methods we will take adaptive Newton
multilevel FEMs, whenever it comes to numerical examples.
In Section 3.4.3 we had already discussed the finite dimensional analogue, the
global inexact Newton-PCG method. With the theoretical considerations at
the beginning of Section 8.3, we are prepared to modify the global convergence
theorems for the Newton-PCG methods in such a way that they apply to the
more general Newton-Galerkin case. In what follows, we just combine and
modify our previous Theorems 3.23 and 3.26.
Theorem 8.7 Notation as introduced above. Let f : D → R1 be a strictly
convex C 2 -functional to be minimized over some open convex domain D ⊂
W 1,p and F (x) = f (x) be strictly positive. For x, y ∈ D, assume the affine
conjugate Lipschitz condition
F (x)−1/2 (F (y) − F (x))(y − x) ≤ ωF (x)1/2 (y − x)2
with 0 ≤ ω < ∞. Let Δxk denote the exact and δxk the inexact Newton
correction. For each well-defined iterate xk ∈ D, define the quantities
k = F (xk )1/2 Δxk 2 ,
δk = F (xk )1/2 δxk 2 =
hk = ωF (xk )1/2 Δxk ,
hδk = ωF (xk )1/2 δxk =
Moreover, let xk + λδxk ∈ D for 0 ≤ λ ≤ λkmax with
k
,
1 + δk2
hk
.
1 + δk2
398
8 PDE Boundary Value Problems
λkmax :=
4
1 + 8hδk /3
1+
≤ 2.
Then
f (xk + λΔxk ) ≤ f (xk ) − tk (λ)δk
where
tk (λ) = λ − 12 λ2 − 16 λ3 hδk .
The optimal choice of damping factor is
λk =
1+
2
1 + 2hδk
≤ 1.
As in the local convergence case, hk is the Kantorovich quantity and δk the
relative Galerkin approximation error.
Adaptive damping and accuracy matching. Following our usual paradigm, the unknown theoretical quantities hk and δk are replaced by computationally available estimates [hk ] and [δk ]. For [hk ] we just use the terms
E1,2,3 (λ) as given in Section 3.4.2 for the exact Newton method and in Section
3.4.3 for the inexact Newton-PCG method. On this basis we realize the correction strategy (3.88) with hk replaced by the a-posteriori estimate [hδk ] and the
prediction strategy (3.89) with hk+1 replaced by the a-priori estimate [hδk+1 ].
Unless stated otherwise, we choose the approximation error bound δk = 1 as
a default throughout the Newton-Galerkin iteration, thus eventually merging
into the linear local convergence mode.
Example 8.8 Good versus bad initial coarse grid. We return to our
previous Example 8.7, but this time for the critical value p = 1, which characterizes the (parametric) minimal surface problem. This value is critical,
since then u ∈ W 1,1 , a nonreflexive Banach space, which implies that the
existence of a unique solution is no longer guaranteed. For special boundary
conditions and inhomogeneities g, however, a unique solution can be shown
to exist, even in C 0,1 (see, e.g., the textbooks by E. Zeidler [205]). Such a
situation occurs, e.g., for
5 π 6 5 π π6
Ω = − ,0 × − ,
, u|∂Ω = s cos x cos y , g ≡ 0 .
2
2 2
Taking the Z2 -symmetry along the x-axis into account, we may halve Ω and
impose homogeneous Neumann boundary conditions at y = 0. The parameter
s is set to s = 3.5. From a quick rough examination of the problem, we expect
a boundary layer at x = 0. As initial guess u0 we take the prescribed values
on the Dirichlet boundary part and otherwise just zero.
Again we solve the problem by Newton-KASKADE. As good initial coarse grid
we select the grid in Figure 8.5, left, which takes the expected boundary layer
into account. As bad initial coarse grid we choose the one in Figure 8.5, right,
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs
399
Fig. 8.5. Example 8.8. Good (left) and bad (right) initial coarse grid.
1
good grid
0.95
0.9
0.85
bad grid
0.8
0.75
0.7
0
2
4
6
8
10
12
14
16
Fig. 8.6. Example 8.8. Comparative damping strategies for good and bad initial
coarse grids.
which deliberately ignores any knowledge about the occurrence of a boundary
layer.
In Figure 8.6, the comparative performance of our global Newton-KASKADE
algorithm is documented in terms of the obtained damping factors for both
initial grids. As expected from reports in the engineering literature, the bad
coarse grid requires many more iterations to eventually capture the nonlinearity.
Example 8.9 Function space versus finite dimensional approach.
Once again, we return to Example 8.7, this time for the critical value p = 1.
In Figure 8.7, we show two settings: On the left (Example 8.9a), a unique
solution exists, which has been computed, but is not documented here; this
400
8 PDE Boundary Value Problems
example serves for comparison only, see Table 8.11 below. Our main interest
focuses on Example 8.9b, where no (physical) solution exists. For the initial
guess u0 we take the prescribed values on the Dirichlet boundary and zero
otherwise.
,u
=0
,n
u=0
,u
=0
,n
u=1
u=cos y
u=0
u=0
,u
=0
,n
,u
=0
,n
u=0
Fig. 8.7. Example 8.9. Domains and initial coarse grids. Black lines: Dirichlet
boundary conditions, grey lines: Neumann boundary conditions. Left: Example 8.9a,
unique solution exists. Right: Example 8.9b, no solution exists.
At Example 8.9b, we want to compare the algorithmic behavior of two different Newton-FEM approaches:
• our function space oriented approach, as presented in this section, and
• the finite dimensional approach, which is typically implemented in classical
Newton-multigrid FEMs.
In the finite dimensional approach, the discrete FE problem is solved successively on each of the mesh levels so that there the damping factors will
repeatedly run up to values λ = 1. In contrast to that behavior, our function
space approach aims at directly solving the continuous problem by exploiting
information available from the whole mesh refinement history. Consequently,
if a unique solution exists, this approach will reach the local convergence
phase in accordance with the mesh refinement process. Such a behavior has
already been shown for our preceding Example 8.8 in Figure 8.6.
Figure 8.8 gives an account for Example 8.9b. In the finite dimensional option, damping factors λ = 1 arise repeatedly on each of the mesh refinement
levels. After more than 60 Newton-FEM iterations, this approach gives the
impression of a unique solvability of the problem—on the basis of the local
convergence of the Newton-FEM algorithm on each of the successive meshes.
Our function space option, however, terminates already after 20 Newton iterations for λ < λmin = 0.01.
8.3 Inexact Newton Multilevel FEM for Elliptic PDEs
401
1
0.1
finite dimensional Newton
function space Newton
0.01
0
10
20
30
40
50
60
Fig. 8.8. Example 8.9b. Iterative damping factors for two Newton-FEM algorithms. To be compared with Figure 8.9.
To understand this discrepancy, we simultaneously look at the local energy
norms k , which measure the exact Newton corrections Δxk , see Figure 8.9.
The finite dimensional method ends up with ‘sufficiently small’ Newton corrections on each of the refinement levels, pretending some local pseudoconvergence. Our function space Newton method, however, stays with ‘moderate
size’ corrections throughout the iteration.
1
function space Newton
0.1
0.01
0.001
0.0001
1e-05
finite dimensional Newton
0
10
20
30
40
1/2
Fig. 8.9. Example 8.9b. Iterative energy norms k
rithms. To be compared with Figure 8.8.
50
60
for two Newton-FEM algo-
402
8 PDE Boundary Value Problems
Asymptotic mesh dependence. Table 8.11 (from [198]) compares the
actually computed affine conjugate Lipschitz estimates [ωj ] as obtained from
Newton-KASKADE in Example 8.9a and in Example 8.9b. Obviously, Example
8.9a, which has a unique solution, exhibits asymptotic mesh independence as
studied in Section 8.1. Things are totally different in Example 8.9b, where
the Lipschitz estimates clearly increase. Note that the blow-up of the lower
bounds [ωj ] in Table 8.11 implies a blow-up of the Lipschitz constants ωj —for
this purpose the bounds are on the rigorous side.
j
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Example 8.9a
unknowns
[ωj ]
4
7
18
50
123
158
278
356
487
632
787
981
1239
1610
2054
1.32
1.17
4.55
6.11
5.25
20.19
19.97
9.69
8.47
11.73
44.21
49.24
20.10
32.93
37.22
Example 8.9b
unknowns
[ωj ]
5
10
17
26
51
87
105
139
196
241
421
523
635
7.5
4.2
7.3
9.6
22.5
50.3
1486.2
2715.6
5178.6
6837.2
12040.2
167636.0
1405910.0
Table 8.11. Computational Lipschitz estimates [ωj ] on levels j. Example
8.9a: unique solution exists, Example 8.9b: no solution exists.
Interpretation. Putting all pieces of available information together, we now
understand that on each of the levels j this problem has a finite dimensional
solution x∗j , unique within the finite dimensional Kantorovich ball with radius
ρj ∼ 1/ωj ; however, these balls shrink from radius ρ1 ∼ 1 down to ρ22 ∼ 10−6 .
Frank extrapolation of this effect suggests that
lim ρj = 0 .
j→∞
Obviously, the algorithm insinuates that a unique continuous solution of the
stated PDE problem does not exist. This feature would certainly be desirable
for any numerical PDE solver.
Exercises
403
Exercises
Exercise 8.1 In Section 8.3.1, a rough computational complexity model of
adaptive multilevel FEM for nonlinear elliptic PDEs leads to the problem
(dropping the index k)
−d/2
I ∼ 1 + 1/δ 2
log 1 + 1/δ 2 = max ,
where d is the spatial dimension.
a) Calculate the maximum point δ and evaluate it for d = 2 and d = 3.
b) How would the rough model need to be changed, if the situation h = 0
were to be modeled?
Exercise 8.2 Consider the finite dimensional Newton sequence
xk+1 = xk + Δxk ,
where x0 is given and Δxk is the solution of a linear system. In sufficiently
large scale computations, rounding errors caused by direct elimination or
truncation errors from iterative linear solvers will generate a different sequence
y k+1 = y k + Δy k + k ,
where y 0 = x0 is given, Δy k is understood to be the exact Newton correction
at y k , and
k ≤ δΔy k ,
Upon using analytical tools of Section 8.1, derive iterative error bounds for
y k − xk and y k − x∗ .
Exercise 8.3
Consider the nonlinear ODE boundary value problem
ẋ = f (x) ,
Ax(a) + Bx(b) = 0
with linear separable boundary conditions. We want to study asymptotic
mesh independence for Gauss collocation methods of order s ≥ 1 (compare
Section 7.4). For the approximating space we select X = W 1,∞ and impose
the assumptions from Section 8.1. Let Xj ⊂ X denote a finite dimensional
subspace characterizing the collocation discretization with maximum mesh
size τj . Assume that f is sufficiently smooth and the BVP is well-conditioned
for all required arguments.
a) In view of (8.5), derive upper bounds δj such that
Δxj − ΔxW 1,∞ ≤ δj
404
8 PDE Boundary Value Problems
with the asymptotic property
δj → 0.
Hint: Compare the exact solution w of
ẇ + fx (xj )w = −fx (xj )xj + f (xj ) ,
Aw(a) + Bw(b) = 0
and its approximation wj using the error estimate (as given in [180, 49])
w − wj W 1,∞ ≤ Cτj ẅ∞ ,
where C is a bounded generic constant, which is independent of j.
b) Under the assumptions of Theorem 2.2 derive some bound
w − wj ≤ σj vj 2
with the asymptotic property
σj → 0 .
Exercise 8.4 We consider linear finite element approximations on quasiuniform triangulations for semilinear elliptic boundary value problems
F (x) = −div∇x − f (x) = 0,
x ∈ H01 (Ω)
on convex polygonal domains Ω ⊂ Rd , d ≤ 3. For this setting, we want to
study asymptotic mesh independence. The notation is as in Section 8.1.
In view of (8.5) and (8.9), derive upper bounds δj such that
Δxj − Δx ≤ δj
and σj such that
w − wj ≤ σj vj 2 ,
Assume that the above right hand term f : R → R is globally Lipschitz continuously differentiable. In particular, show that for the process of successive
refinement the asymptotic properties
lim δj = 0 ,
j→∞
lim σj = 0
j→∞
hold.
Hint: Exploit the H 2 -regularity of xj + Δx and use the embedding H 1 "→ L4 .
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Software
Software
This monograph presents a scheme to construct adaptive Newton-type algorithms in close connection with an associated affine invariant convergence
analysis. Part of these algorithms are presented as informal programs in the
text. Some, but not all of the described algorithms have been worked out in
detail. Below follows a list of codes mentioned by name in the book, which
can be downloaded via the web address
http://www.zib.de/Numerics/NewtonLib/
All of the there available programs (not only by the author and his group)
are free as long as they are exclusively used for research or teaching purposes.
Iterative methods for large systems of linear equations:
• PCG – adaptive preconditioned conjugate gradient method for linear systems with symmetric positive definite matrix; energy error norm based
truncation criterion (Section 1.4.2)
• GBIT – adaptive Broyden’s ‘good’ rank-1 update method specialized for
linear equations; error oriented truncation criterion (Section 1.4.4)
Exact global Newton methods for systems of nonlinear equations:
• NLEQ1 – popular production code; global Newton method with error oriented convergence criterion; arbitrary selection of direct linear equation
solver; adaptive damping strategies slightly different from Section 3.3.3; no
rank strategy
• NLEQ2 – production code; global Newton method with error oriented convergence criterion; QR-decomposition with subcondition number estimate;
adaptive damping and rank strategy slightly different from Section 3.3.3
• NLEQ-RES – global Newton method with residual based convergence criterion and adaptive trust region strategy (Section 3.2.2)
• NLEQ-ERR – global Newton method with error oriented convergence criterion and adaptive trust region strategy (Section 3.3.3)
• NLEQ-OPT – global Newton method for gradient systems originating from
convex optimization; energy error norm oriented or objective function based
convergence criteria and adaptive trust region strategy (Section 3.4.2)
Local quasi-Newton methods for systems of nonlinear equations:
• QNERR – recursive implementation of Broyden’s ‘good’ rank-1 update
method; error oriented convergence criterion (Section 2.1.4)
• QNRES – recursive implementation of Broyden’s ‘bad’ rank-1 update method;
residual based convergence criterion (Section 2.2.3)
Software
417
Continuation methods for parameter dependent systems of nonlinear equations:
• CONTI1 – global Newton continuation method (classical and tangent continuation); no path-following beyond turning points (Section 5.1.3)
• ALCON1 – global quasi-Gauss-Newton continuation method; adaptive pathfollowing beyond turning points (Section 5.2.3)
• ALCON2 – global quasi-Gauss-Newton continuation method; adaptive pathfollowing beyond turning points; computation of bifurcation diagrams including simple bifurcations (Sections 5.2.3, 5.3.2, and 5.3.3)
Global Gauss-Newton methods for nonlinear least squares problems:
• NLSCON – (older) global constrained (or unconstrained) Gauss-Newton
method with error oriented convergence criterion; adaptive trust region
strategies slightly different from Sections 4.3.4 and 4.1.2
• NLSQ-RES – global unconstrained Gauss-Newton method with projected
residual based convergence criterion and adaptive trust region strategy
(Section 4.2.3)
• NLSQ-ERR – global unconstrained Gauss-Newton method with error oriented convergence criterion and adaptive trust region strategies (Sections
4.3.4 and 4.3.5)
Inexact global Newton methods for large systems of nonlinear equations:
• GIANT – (older) global inexact Newton method with error oriented convergence criterion; adaptive trust region strategy slightly different from
Sections 2.1.5 and 3.3.4; earlier version of GBIT for inner iteration
• GIANT-GMRES – global inexact Newton method with residual based convergence criterion and adaptive trust region strategy; GMRES for inner iteration
(Sections 2.2.4 and 3.2.3)
• GIANT-GBIT – global inexact Newton method with error oriented convergence criterion and adaptive trust region strategy; GBIT for inner iteration
(Sections 2.1.5 and 3.3.4)
• GIANT-PCG – global inexact Newton method for gradient systems originating from convex function optimization; energy error norm oriented or
function based convergence criteria and adaptive trust region strategy; PCG
for inner iteration (Sections 2.3.3 and 3.4.3)
418
Software
Multiple shooting methods for ODE boundary value problems:
• BVPSOL – Multiple shooting method for two-point boundary value problems; exact global Newton method with error oriented convergence criterion and adaptive trust region strategies (Section 7.1.2)
• MULCON – Multiple shooting method for two-point boundary value problems; adaptive Gauss-Newton continuation method (Section 7.1.3)
• PERIOD – multiple shooting method for periodic solutions of ODEs; global
underdetermined Gauss-Newton method with error oriented convergence
criterion and adaptive trust region strategies (Section 7.3.1)
• PERHOM – multiple shooting method for periodic solutions of parameter
dependent ODEs; adaptive error oriented underdetermined Gauss-Newton
orbit continuation method (Section 7.3.2)
• PARKIN – single shooting method for parameter identification in large reaction kinetic ODE networks; global Gauss-Newton method with error oriented convergence measure and adaptive trust region strategies (Section
7.2)
Adaptive multilevel finite element methods for elliptic PDEs:
• KASKADE – function space oriented additive multilevel FEM for linear elliptic PDEs; hierarchical basis preconditioning in 2D; BPX preconditioning
in 3D (Section 1.4.5)
• Newton-KASKADE – function space oriented global inexact Newton multilevel FEM for nonlinear elliptic PDEs originating from convex optimization;
energy error norm oriented or objective functional based convergence criteria and adaptive trust region strategy; adaptive multilevel code KASKADE
for inner iteration (Section 8.3); this code is still in the form of a research
code which is not appropriate for public distribution (see above web address where possible cooperation is discussed)
Index
Abdulle, A., 173
active set strategy, 204
affine conjugacy, 16
affine contravariance, 15
affine covariance, 14
affine similarity, 17
Aigner, M., 394
aircraft stability problem, 262
Akhilov, G., 48
ALCON1, 236, 262, 417
ALCON2, 279, 280, 417
Allgower, E.L., 377
Apostolescu, V., 205
arclength continuation, 251, 280
Armbruster, D., 271
Armijo strategy, 121, 129, 146
Ascher, U.M., 142, 316, 350
asymmetric cusp, 269
asymptotic distance function, 139, 212
asymptotic mesh independence
– collocation method, 363, 403
– finite element method, 388, 401, 404
attraction basins, 151
AUTO, 251, 332
automatic differentiation, 24, 312
Axelsson, O., 26, 28, 145
Böhmer, K., 377
Babuška, I., 40, 361, 395
Bader, G., 48, 321, 327, 350
Bank, R.E., 38, 40, 94, 134
Bastian, P., 39, 395
BB, 107
Ben-Israel, A., 175
BFGS rank-2 update formula, 105, 106
Bi-CG, 27
Bi-CGSTAB, 27
bifurcation
– diagram, 234
– imperfect, 273
– perfect, 273
– simple, 267
– simple point, 234
bit counting lemma, 51, 129, 146, 157,
165, 191, 213
Bjørck, Å., 173, 175, 200, 336
block Gaussian elimination, 335
Blue, J., 220
Bock, H.G., 15, 21, 48, 142, 201, 205,
241, 316, 324, 330, 336
Boggs, P.T., 185
Bornemann, F., 39, 40
boundary conditions
– separable linear, 320
BOUNDSCO, 327
Braess, D., 188
Brezillon, J., 305
Broyden, C.G., 66, 81
Bulirsch, R., 318, 328
Burkardt, J.V., 250
BVP condition number, 352
BVP Green’s function, 352
BVPSOL, 327, 418
Cauchy, A., 114
CCG, 39
CGNE, 7, 27, 32, 34, 35, 38, 67–69, 73–76,
152, 155–161, 383
CGNR, 28
chemical reaction networks, 336
chemical reaction problem, 261
Christiansen, J., 350
COCON, 350
COLCON, 350
collocation conditions
– Fourier, 348
419
420
Index
– global, 349
– local, 349
collocation method
– asymptotic mesh independence, 363,
403
– error estimation, 358
– error matching, 363
– ghost solutions, 356
– global mesh selection, 360
– inconsistent discrete solutions, 355
– spurious solutions, 356
– superconvergence, 351, 355
COLNEW, 350
COLSYS, 350, 360, 361, 363
condition number estimate
– by iterative refinement, 249, 322
Conn, A.R., 117
CONTI1, 245, 417
continuation method
– classical, 237, 238, 327
– – nonlinear least squares problems,
281
– discrete, 235
– Euler, 237
– explicit reparametrization, 250
– of incremental load, 237
– order, 238, 241
– pseudo-transient, 111
– tangent, 237, 239, 327
– – nonlinear least squares problems,
281
– trivial BVP, 328
– turning angle restriction, 257
convergence monitor
– energy norms, 98
– functional values, 96
– inexact energy norms, 102
– inexact Newton corrections, 73
– ordinary Newton corrections, 51
– quasi-Newton corrections, 61
– residual norms, 78, 81, 92
– simplified Newton corrections, 55
Crandall, M.G., 267
critical point, 143
– detection, 249, 259, 332
– higher order, 270
cyclic block matrix, 317
cyclic nonlinear system, 317
Dahlquist stability model, 296
Dahlquist, G., 289, 296
DASSL, 150
Davidenko differential equation, 233
Davidenko, D., 125, 233
Dellnitz, M., 344
Dembo, R.S., 94, 134
Dennis, J.E., 15, 66, 119, 185
DFP rank-2 update formula, 105, 106
Dickmanns, E.D., 144, 328, 329
discrete L2 -product, 37, 325, 332
discrete PDEs, 378
Doedel, E., 251, 332
downhill property, 115
driven cavity problem, 379
dynamical invariant, 299, 313
Eisenstat, S.C., 94, 134
elastomechanics problem, 385
enzyme reaction problem, 200
equivariant nonlinear systems, 279
Erdmann, B., 39, 40
extrapolation
– local, 362
fast Givens rotation, 230
Fiedler, B., 262, 279, 332
Fischer, B., 31, 34
fixed point iteration, 300
Fletcher, R., 104
Fourier collocation method
– adaptivity, 347, 367
– collocation conditions, 348
Fourier, J., 10
Freund, R., 35
Galerkin condition, 31, 33, 168, 391
Gatermann, K., 279, 340, 344
Gauger, N., 305
Gauss collocation, 350
Gauss, Carl Friedrich, 173, 177
Gauss-Newton method, 25
– ‘simplified’, 197
– constrained, 202, 335
– separable, 187, 202, 231
– unconstrained, 185, 197
Gauss-Newton path
– geodetic, 225, 227
– global, 205
Index
– local, 205
Gay, D.M., 38, 119
GB, 36
GBIT, 7, 25, 27, 35, 36, 38, 41, 67, 70,
73–76, 152, 156–161, 364, 383, 384,
416, 417
Georg, K., 236
GIANT, 23, 76, 161, 381, 417
GIANT-CGNE, 24, 76, 160, 161
GIANT-CGNE/L, 382, 383
GIANT-GBIT, 24, 76, 160, 161, 417
GIANT-GBIT/L, 382–384
GIANT-GMRES, 24, 94, 133, 161, 417
GIANT-GMRES/L, 381, 382
GIANT-GMRES/R, 381, 382
GIANT-PCG, 24, 104, 169, 394, 417
Gill, P.E., 185
Glowinski, R., 104
GMRES, 7, 27–29, 35, 90, 91, 93, 94, 107,
126, 131–133, 172, 307, 309, 310, 312,
314, 417
Golub, G.H., 23, 31, 34, 175, 188, 200,
202
Golubitsky, M., 235, 267
Gould, N.I.M., 117
greedy algorithm, 394
Greville, T.N.E., 175
Griewank, A., 22, 24, 175, 312
Hölder continuity, 15, 105
Hackbusch, W., 26, 38, 369
Hairer, E., 289, 373
harmonic balance method, 367
Hazra, S.B., 305
Healey, T.J., 280
Hebden, M.D., 119
Heindl, G., 15, 48, 205
hexagonal lattice dome problem, 280
highly nonlinear, 109
Hirsch, M.W., 241
Hohmann, A., 16, 279, 340, 344, 350,
359, 363
homotopy, 111, 288
– discrete, 111
– multipoint, 170
homotopy path, 233, 300
– tangent computation
– – via LU-decomposition, 252
– – via QR-decomposition, 252
421
Hopf bifurcation points, 344
Householder, A.S., 60
HYBRJ1, 151
implicit Euler discretization, 313
implicit function theorem
– affine covariant, 40
implicit midpoint rule, 298
incompatibility factor, 198
– constrained nonlinear least squares
problem, 204
inexact Newton method
– computational complexity model, 393
– function space, 356, 363
– – computational complexity model,
403
inner inverse, 176, 182
internal differentiation, 324
iterative refinement sweeps, 321, 336
Jankowsky, I., 249
Jennrich, R.L., 202
Jepson, A.D., 344
Jordan canonical form, 18
Kantorovich, L.V., 46, 48
KASKADE, 39, 395, 396, 418
Kaufman, L., 188, 202
Keller, H.B., 15, 48, 105, 241, 251
Kelley, C.T., 26, 66, 305
Keyes, D.E., 305
Kollerstrom, N., 9
Kornhuber, R., 26, 39, 40, 369
Kostina, E.A., 142
Kowalik, J., 200
Krylov subspace, 29, 33, 93
Kubiček, M., 261
Kunkel, P., 262, 270, 279, 332, 350
Kuznetsov, Y.A., 30
Lahaye, E., 235, 238
LAPACK, 23
Leinen, P., 39
level function
– general, 116, 121, 135
– hybrid, 326, 367
– natural, 326, 367
– – Gauss-Newton method, 212
– residual, 114, 126, 326, 367
422
Index
level set
– general, 121
– residual, 114, 126
Levenberg, K.A., 118
Levenberg-Marquardt method, 110, 118
LIMEX, 150, 151, 313
Lindström, P., 193
linear contractivity, 301
– of discretizations, 296
linear convergence mode
– Newton-ERR, 75
– Newton-PCG, 103
– Newton-RES, 93
linear least squares problem
– equality constrained, 178
– unconstrained, 175
linearly implicit discretizations, 296
linearly implicit Euler discretization,
300, 312
Liniger, W., 290
Lobatto collocation, 350
local discretization error
– equidistribution, 360
local extrapolation, 361
Lyapunov-Schmidt reduction, 263
– perturbed, 269, 270
M-matrix property, 112
Marchuk, G.I., 30
Marquardt, D.W., 118
Mars satellite orbit problem, 144
Matlab, 249
maximum likelihood method, 173
McCormick, S.F., 377
Melhem, R.G., 262
Menzel, R., 241, 269
mesh equilibration, 395
mesh refinement
– global, 362
– local, 362
Meurant, G., 31, 34
mildly nonlinear, 79, 109
minimal surface problem
– convex domain, 385
– nonconvex domain, 386
MINPACK, 151
monotonicity test
– functional
– – restricted, 166
– natural, 110, 134, 138
– – Gauss-Newton method, 213
– – inexact, 159
– – restricted, 146
– – underdetermined Gauss-Newton
method, 227
– residual, 110
– – algorithmic limitation, 137
– – Gauss-Newton method, 192
Moore, G., 263, 267, 271, 332
Moore-Penrose pseudo-inverse, 176,
222, 323, 332
Moré, J.J., 66, 118, 119
Morgan, A.P., 241
MULCON, 332, 418
multilevel Newton method, 26, 369
Murray, W., 185
Mysovskikh, I., 48
Nashed, M.Z., 175
Newton method
– algebraic derivation, 8
– exact, 22
– geometrical derivation, 9
– historical note, 9
– in function space, 25
– inexact, 23
– – matrix-free, 24
– ordinary, 22
– – computational complexity, 79
– scaling effect, 13
– scaling invariance, 19
– simplified, 22, 245
– truncated, 23, 94
Newton multigrid method, 26
Newton multilevel method, 26
Newton path, 122, 161
– computation, 150
– inexact, 132, 154
Newton, Isaac, 9
Newton-Kantorovich theorem
– affine contravariant, 79, 281
– affine covariant, 40, 46, 243
– classical, 12, 40, 48
Newton-KASKADE, 395, 398, 399, 402,
418
Newton-like method, 22, 104
Newton-Mysovskikh theorem
– affine conjugate, 94
Index
– affine contravariant, 77, 231, 281
– – for Gauss-Newton method, 183
– affine covariant, 48
– – for Gauss-Newton method, 194
– classical, 12, 48, 104
– refined, 50
Newton-Raphson method, 11
Newton-Simpson method, 11
NITSOL, 134
NLEQ, 23, 380
NLEQ-ERR, 23, 66, 148, 161, 380–383,
416
NLEQ-ERR∗ , 383
NLEQ-ERR/I, 383
NLEQ-OPT, 23, 416
NLEQ-RES, 23, 131, 380–382, 416
NLEQ-RES/L, 380–382
NLEQ1, 60, 66, 147, 148, 151, 416
NLEQ2, 416
NLSCON, 220, 417
NLSQ-ERR, 220, 417
NLSQ-RES, 193, 417
nonlinear CG, 104
nonlinear contractivity, 293, 304
– of discretizations, 296
– of implicit Euler discretization, 296
– of implicit midpoint rule, 298
– of implicit trapezoidal rule, 298
nonlinear least squares problem
– ‘small residual’, 185
– ‘wrong’, 197, 229
– adequate, 198
– compatible, 174
– constrained, 333, 335
– inadequate, 215
– incompatible, 183
– separable, 187, 202
nonlinear preconditioning, 140, 161
normal equations, 177
Nowak, U., 76, 161, 336, 378
numerical rank decision, 177, 180
Oberle, J., 327
optimal line search, 116
OPTSOL, 328, 329
Ortega trick, 54, 81
Ortega, J.M., 26, 52, 112, 114
orthogonal projectors, 176
Osborne, M.R., 142, 200, 202
423
outer inverse, 176, 194, 323, 326
PARFIT, 336
PARKIN, 336, 418
PCG, 7, 27, 30, 31, 98, 99, 102, 104, 168,
169, 391, 393, 394, 397, 398, 416, 417
Peano theorem, 288
penalty limit method, 179
penalty method, 179
Penrose axioms, 176, 231, 367
Pereyra, V., 185, 188, 202
PERHOM, 340, 344, 418
PERIOD, 340, 418
periodic orbit computation
– gradient method, 366
– multiple shooting, 338
– simplified Gauss-Newton method,
340
– single shooting, 338
periodic orbit continuation
– multiple shooting, 342
– single shooting, 341
Pernice, M., 134
Pesch, H.-J., 329
Petzold, L., 150
PGBIT, 38
Picard iteration, 288
PITCON, 250
PLTMG, 39
Poincaré, H., 235, 238
Polak, E., 104
polynomial continuation
– feasible stepsize, 241
– stepsize control, 248
Potra, F.A., 48, 114, 377
preconditioning, 73, 93, 389
– nonlinear, 140, 161
– of dynamical systems, 312
prediction path, 237
propagation matrix, 352, 368
PSB rank-2 update formula, 106
pseudo-arclength continuation, 251, 280
pseudo-timestep strategy, 305, 310
Pták, V., 48
QNERR, 61, 148, 149, 224, 245, 253, 416
QNRES, 89, 130, 131, 416
QR-Cholesky decomposition, 178, 222,
225, 230
424
Index
quadratic convergence mode
– Newton-ERR, 75
– Newton-PCG, 103
– Newton-RES, 92
quasi-Gauss-Newton method, 224
quasi-Newton method, 25, 245
– in multiple shooting, 325
quasilinearization, 25, 370
Rabinowitz, P.H., 264, 267
Rall, L.B., 48, 105
– theorem of, 105
rank strategy
– in multiple shooting, 326
Rannacher, R., 396
Raphson, Joseph, 10
Reeves, C.M., 104
Rheinboldt, W.C., 26, 40, 48, 52, 105,
112, 114, 250, 262, 361, 377
Ribière, R., 104
Rose, D.J., 94, 134
Ruhe, A., 188
Russell, R.D., 316, 350
Saad, Y., 28, 30
Schaeffer, D., 267
Schiela, A., 378
Schlöder, J.P., 142
Schnabel, R.B., 38, 66
Schultz, M.H., 30
Schulz, V., 305
Schwetlick, H., 241, 261
scoring method, 25, 202
secant condition, 25, 325
– affine contravariant, 82
– affine covariant, 58
sensitivity matrices, 334
Shaidurov, V., 39
Sherman-Morrison formula, 36, 60, 83
simplified Newton correction, 138
– inexact, 154
simplified Newton method
– stiff initial value problems, 290
Simpson, Thomas, 10
Smale, S., 241
small residual factor, 185, 191
small residual problem, 185
space shuttle problem, 328
SPARSPAK, 23
standard embedding, 239
– partial, 239, 245
steepest descent, 114
– Gauss-Newton method, 212
– method, 110, 229
– Newton method, 139
Steihaug, T., 94, 134
Stoer, J., 205, 318
subcondition number, 178, 249, 323,
328
supersonic transport problem, 379
SYMCON, 279, 280
Toint, P.L., 117
truncation index, 23
turning point, 234, 250
– computation, 259
UG, 39, 395
universal unfolding, 266
Urabe, M., 345
van der Vorst, H.A., 27
van Loan, C.F., 23, 175
Varah, J.M., 357
Varga, R.S., 112
Vieta, Francois, 9
Volkwein, S., 20
Walker, H.F., 66, 94, 134
Walter, A., 35
Walter, W., 293
Wanner, G., 173, 289, 373
Wedin, P.-Å., 185, 188, 193, 229
Weiser, M., 17, 20, 104, 170, 350, 366,
378
Welsch, R., 119
Werner, B., 344
Wilkinson, J.H., 200
Willoughby, R.A., 290
Wittum, G., 39, 395
Woźniakowski, H., 249
Wronskian matrix, 352
Wulff, C., 340, 344
X-ray spectrum problem, 220
Xu, J., 39
Yamamoto, T., 15, 46, 48
Ypma, T.J., 9, 15, 76, 161
Yserentant, H., 39, 395
Zeidler, E., 398
Zenger, C., 178